FAQ: How to design an anti-aliasing architecture using Sallen-Key filters

Introduction

This KWIK (Know-How and Knowledge Integration) circuit application note provides a step-by-step guide to solving specific design challenges. Given a set of application circuit requirements, this article shows how to address these requirements using general formulas and make them easily scalable to other similar application specifications.

In any sampling system, such as a measurement system involving an ADC, there is a phenomenon called aliasing that can cause signals in a higher frequency band to "fold down" into the Nyquist band, making them indistinguishable from the signal of interest. The Nyquist frequency is half the sampling rate, fs. The bandwidth of the circuit sampled by the ADC should be less than half the sampling rate. Aliasing can cause interfering signals and noise to contaminate the output, affecting measurement accuracy. Figures 1 and 2 show examples of correct sampling (high sampling rate) and incorrect sampling (low sampling rate), respectively.

Figure 1: The sampling rate is high enough to resolve the signal

Figure 2: Low sampling rate, aliasing back to DC

Aliasing is a particular problem for measurement systems used in applications that rely on data analysis in the frequency domain, such as condition monitoring, preventive maintenance, power quality monitoring, and passive sonar. These systems typically support wide bandwidth data acquisition, are able to process signals such as vibration, power, and acoustics in the frequency domain, and make decisions based on the characteristics of signal tones and harmonics. The nature of the monitoring requirements makes them particularly sensitive to spectral interference. Low-pass filters are an important part of the signal chain design in these measurement systems, helping to prevent out-of-band signals and noise from folding into the signal bandwidth of interest.

Active filters are often chosen over passive filters for signal processing circuits because they offer output power gain and frequency range that can be easily adjusted by modifying the filter parameters. Butterworth, elliptic, Bessel, and Chebyshev filters are four of the most common active filters. Choosing the most appropriate filter really depends on the application and the trade-offs that need to be made in the filter design process, such as frequency response, order, or phase linearity. Figure 3 shows the response of each active filter, and the design tips section provides further explanation.

Figure 3: Four types of active filters

The choice here is to implement a Butterworth filter. Once the filter type is chosen, the configurations commonly used to support active pole-pair topologies are Sallen-Key filters and multiple feedback filters. This KWIK circuit note will help design and implement a low-pass Sallen-Key filter to prevent aliasing.

A single-stage Sallen-Key filter consists of an active device (op amp) and passive components (resistors and capacitors). The voltage gain of the op amp is set by a two-resistor divider, and the op amp is used to reject out-of-range signals. This article discusses an example filter circuit design for a low power application. The principles discussed apply to general filter design. The design uses the LTC2066, a 10µA supply current, low IB, zero-drift op amp. There are other op amps in our low power signal chain portfolio that can provide similar results (AD8505, LTC6258, LTC2063, ADA4505, MAX40023, MAX40108).

Figure 4 shows the overall schematic of the circuit, where the LTC2066 is chosen to implement a Sallen-Key, two-pole, unity-gain, low-pass filter.

Design Specification Example

Low-power applications typically use a single supply, typically between 1.8 V and 3.6 V. The design choices for the circuit shown in Figure 4 depend on the sensor output and the application requirements. For this case, one possible sensor is the ADLX356B, an accelerometer used to measure vibrations in machinery in very noisy environments with many high-frequency interference signals. An example of the key design specifications for the circuit in Figure 4 is listed in Table 1.

Figure 4: Sallen-Key low-pass filter with unity gain

Table 1. Key design specifications for the circuit shown in Figure 4

Design Description

The circuit in Figure 4 includes the LTC2066, a 10µA supply current, low IB, zero-drift op amp that can operate from a minimum 3.3V supply, VDD. The LTC2066 is available in a 6-lead SC70 package. This low-pass Sallen-Key filter is used to filter the output of the accelerometer to prevent aliasing into the frequency band of interest.

The output of the filter is denoted as Vout, while Vsignal represents the output of the accelerometer.

Design Considerations

1. The circuit in Figure 4 is a unity-gain Sallen-Key filter. A resistor network can be added to the feedback loop of the op amp to add gain to the circuit (see Appendix 1). The quality factor Q (a measure of performance in terms of stability) is inherently related to gain. For best stability, the gain should be less than 3.

2. The quality factor Q can be important for Sallen-Key filters. In this design, Q is kept at √2⁄2, which is a standard value for a Butterworth second-order filter. Q can be adjusted to change the filter response. The higher the Q (Chebyshev filter), the faster the roll-off, but the phase linearity and amplitude flatness will be reduced. The lower the Q (Bessel filter), the better the phase linearity, but the rejection performance will be reduced. Other parameters that are important in describing low-pass filters can be seen in the labels on Figure 5.

Figure 5: Typical amplitude-frequency behavior of a low-pass filter

3. Unlike an ideal low-pass filter response, the response of the Sallen-Key filter does not continue to decrease beyond the cutoff frequency. After the cutoff frequency, the filter gain will decrease at a slope of 40dB/decade until a certain frequency (depending on the amplifier selected), and then the filter gain begins to increase at a slope of 20dB/decade until it becomes a constant value. This behavior stems from the fact that op amps are not ideal, have a finite gain bandwidth, and their output impedance is not zero. This finite gain bandwidth causes the op amp to lose its vitality at frequencies that exceed its capabilities. When this happens, the op amp looks like an output impedance connected to ground because capacitors C1 and C2 behave as a short circuit.

Not only should the high frequency characteristics of the Sallen-Key filter be monitored, but it is also important to know at what frequency it begins to improve again, as shown by the green curve in Figure 6, while the blue curve is the ideal characteristic. This limit is called the stopband limit.

Figure 6: Stopband limitation

4. In a Sallen-Key filter, the values ​​chosen for the resistor and capacitor are inversely proportional, meaning that if the resistor is larger, then the capacitor will be smaller.

When the sensor does not drive the filter well, it is useful to use a large resistor and a small capacitor so that the sensor is not burdened. Finally, the Sallen-Key filter introduces a problem, as described in Design Tip 3, in that its ability to reject signals decreases at high frequencies. Larger resistors can alleviate this problem.

Smaller resistors and larger capacitors also have some benefits. Smaller resistors are good for DC performance. Input bias current flows into the op amp inputs, so lower resistor values ​​result in lower DC offsets. This combination also reduces noise. Finally, at higher frequencies, smaller resistors reduce the effects of parasitic capacitance on the design.

For this low power design, it is better to use large resistors and small capacitors.

5. When selecting passive component values ​​(resistors and capacitors), tolerances should be considered.

For example, when selecting the values ​​of R1, R2, C1, and C2, the tolerances of these components may change the filter characteristics, such as the cutoff frequency. In a multichannel solution, component tolerances can also affect the matching between channels.

Design Steps

1. Set the filter cutoff frequency:

Before beginning to design a Sallen-Key filter, the sensor specifications must be considered. For this design, the bandwidth of interest for the accelerometer is 3kHz, so the target cutoff frequency for the Sallen-Key filter design will be set to fc = 3kHz.

Note: The formula for calculating the cutoff frequency based on the low-pass Sallen-Key transfer function is:

2. Set the filter’s Q:

There are a number of ways to implement a Sallen-Key filter and evaluate the values ​​of the components. A common approach that does not constrain the design, without implementing gain, is to set the two resistors and two capacitors using ratios. That is, resistors R1 and R2 are set as follows:

The capacitors C1 and C2 are set as follows:

The first step is to choose the values ​​of m and n. From Design Tip 2, we know that the quality factor Q = √2⁄2 ≈ 0.707, but because the gain K is equal to 1, we can also simplify the expression for the Q factor from the transfer function:

Since there are more unknown parameters than known parameters, a ratio is usually fixed. A common value is n = 10, which is what this design procedure will use. Since n = 10, we can now find the value of m.

The complete solution of this quadratic equation is given in the Appendix. The two solutions for m are:

Solving for m1 and m2 are two values ​​for which we can obtain a quality factor of √2⁄2.

3. Component values:

From Design Considerations #4, the expectation is to use large resistors and small capacitors within reasonable limits. Now that the ratio is a known value, the next step is to choose a value for R1 to calculate the other component values. A reasonable high value is R1 = 51kΩ, which is a standard resistor value. This value is also high enough so that the output impedance of the low power op amp LTC2066 is not too large. The previous step yielded two solutions for the resistor ratio. In the Sallen-Key filter, it is better to put the low value resistor in the R1 position for impedance reasons, so the m1 solution will be retained.