Active Low-Pass Filter Design

Abstract

This report focuses on active low-pass filter design using operational amplifiers. Low-pass filters are commonly used to implement anti-aliasing filters in data acquisition systems. Design of second-order filters is the main topic of consideration.

Filter tables are developed to simplify circuit design based on the idea of cascading lower-order stages to realize higher-order filters. The tables contain scaling factors ( $F S F$ $F S F$ ) for the corner frequency ( $f_{c}$ $f_{c}$ ) and the required quality factor ( $Q$ $Q$ ) of each of the stages for the particular filter being designed. The tables enable designers to skip straight to calculating required circuit component values.

To illustrate an actual circuit implementation, six circuits, separated into three types of filters (Bessel, Butterworth, and Chebyshev) and two filter configurations (Sallen-Key and Multiple Feedback), are simulated using a TLV9062 operational amplifier. Limiting factors in the high-frequency performance of the filters are also examined. The tables in this document are used in the Analog Engineer’s Circuit Cookbook: Amplifiers.

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1 Introduction

There are many books that provide information on popular filter types like the Butterworth, Bessel, and Chebyshev filters. This application note examines how to implement these three types of filters.

The mathematics used to transform standard filter table data into the transfer functions required to build filter circuits is examined. Using the same method, filter tables are developed that enable the designer to skip straight to the calculation of the required circuit component values. Actual filter implementation is shown for two circuit topologies: the Sallen-Key and the Multiple Feedback (MFB). The Sallen-Key circuit is sometimes referred to as a voltage-controlled voltage source (VCVS) filter.

Circuits are often referred to as Butterworth filters, Bessel filters, or a Chebyshev filters because their transfer function has the same coefficients as the Butterworth, Bessel, or the Chebyshev polynomial. The MFB or Sallen-Key circuits are also often referred to as filters. The difference is that the Butterworth filter defines a transfer function that can be realized by many different circuit topologies (both active and passive), while the MFB or Sallen-Key circuit defines an architecture or a circuit topology that can be used to realize various second-order transfer functions.

The choice of circuit topology depends on performance requirements. The MFB is generally preferred because the MFB has better sensitivity to component variations and better high-frequency behavior. The unity-gain Sallen-Key inherently has the best gain accuracy because the gain is not dependent on component values.

Table 1-1 and Table 1-2 give a brief summary of the overall trade-offs.

Table 1-1 Summary of Filter Type Trade-Offs

Filter Type	Advantages	Disadvantages
Butterworth	Maximum pass-band flatness	Slight overshoot in response to pulse input and moderate rate of attenuation above $f_{c}$ $f_{c}$
Bessel	Constant group delay – no overshoot with pulse input	Slow rate of attenuation above $f_{c}$ $f_{c}$
3-dB Chebyshev	Fast rate of attenuation above $f_{c}$ $f_{c}$	Large overshoot and ringing in response to pulse input

Table 1-2 Summary of Architecture Trade-Offs

Architecture	Advantages	Disadvantages
Sallen-Key	Not sensitive to component variation at unity gain	High-frequency response limited by the frequency response of the amplifier
MFB	Less sensitive to component variations and excellent high- frequency response	Less simplifications available to ease design