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.

If an ideal low-pass filter existed, it would completely eliminate signals above the cutoff frequency and perfectly pass signals below the cutoff frequency. In real filters, various trade-offs are made to get optimum performance for a given application.

Butterworth filters are termed maximally-flat-magnitude-response filters, optimized for gain flatness in the pass band. The attenuation is –3 dB at the cutoff frequency. Above the cutoff frequency, the attenuation is –20 dB/decade/order. The transient response of a Butterworth filter to a pulse input shows moderate overshoot and ringing.

Bessel filters are optimized for maximally-flat time delay (or constant-group delay). This means that they have linear phase response and excellent transient response to a pulse input. This comes at the expense of flatness in the pass-band and rate of rolloff. The cutoff frequency is defined as the –3 dB point.

Chebyshev filters are designed to have ripple in the pass band, but steeper rolloff after the cutoff frequency. Cutoff frequency is defined as the frequency at which the response falls below the ripple band. For a given filter order, a steeper cutoff can be achieved by allowing more pass-band ripple. The transient response of a Chebyshev filter to a pulse input shows more overshoot and ringing than a Butterworth filter.

When constructing a filter, there are two topologies that can be used: the Sallen-Key topology, which is a non-inverting circuit, or the Multiple Feedback (MFB) topology, which creates an inverting second-order stage circuit. See the Filter Designer tool or the filtering cookbooks for more information on the Sallen-Key and MFB filters.

The transfer function ${\mathrm{H}}_{\mathrm{LP}}$ of a second-order low-pass filter can be expressed as a function of frequency ( $f$ ) as shown in Equation 17, the Second-Order Low-Pass Filter Standard Form.

In this equation, $f$ is the frequency variable, $f_{\mathrm{c}}$ is the cutoff frequency, $FSF$ is the frequency scaling factor, and $Q$ is the quality factor. Equation 17 has three regions of operation: below cutoff, in the area of cutoff, and above cutoff. For each area, Equation 17 reduces to:

• $f\ll f_{\mathrm{c}}\Rightarrow {\mathrm{H}}_{\mathrm{LP}}\left(f\right)\approx K$
  • The circuit passes signals multiplied by the gain factor $K$ .
• $\frac{f}{f_{\mathrm{c}}}=FSF\Rightarrow {\mathrm{H}}_{\mathrm{LP}}\left(f\right)=jKQ$
  • Signals are phase-shifted 90° and modified by the $Q$ factor.
• $f\gg f_{\mathrm{c}}\Rightarrow {\mathrm{H}}_{\mathrm{LP}}\left(f\right)\approx -K{\left(\frac{FSF\times f_{\mathrm{c}}}{f}\right)}^2$
  • Signals are phase-shifted 180° and attenuated by the square of the frequency ratio.

With attenuation at frequencies above $f_{\mathrm{c}}$ increasing by a power of two, the last formula describes a second-order low-pass filter.

The frequency scaling factor $FSF$ is used to scale the cutoff frequency of the filter so that it follows the definitions given before.