- Sallen–Key filter
A Sallen–Key filter is a type of
active filter , particularly valued for its simplicity. The circuit produces a 2-pole (12 dB/octave)lowpass orhighpass response using two resistors, two capacitors and (usually) a unity-gain buffer amplifier . Higher-order filters can be obtained by cascading two or more stages. This filter topology is also known as avoltage controlledvoltage source (VCVS) filter. It was introduced byR.P. Sallen andE. L. Key ofMIT Lincoln Laboratory in 1955.cite journal|title=A Practical Method of Designing RC Active Filters|journal=IRE Transactions on Circuit Theory|date=1955-03|first=R. P.|last=Sallen|coauthors=E. L. Key|volume=2|issue=1|pages=74–85|id= |url=|format=|accessdate=2007-08-14]Although the filters depicted here have a passband
gain of 1 (or 0 dB), not all Sallen–Key filters have a gain of 1 in the passband. Non-unity-gain buffers can also be used (e.g. , by adding additional resistors to theoperational amplifier to change thefeedback gain, as in a non-inverting amplifier with gain greater than 1). Sallen–Key filters are relatively resilient to component tolerance, although obtaining highQ factor may require extreme component values (or higher buffer gain).Generic configuration
A generic unity-gain Sallen–Key filter, implemented with a unity gain
operational amplifier , is shown below with an analysis that uses idealoperational amplifier theory:If the component was connected to ground, the filter would be a
voltage divider composed of the and components cascaded with another voltage divider composed of the and components. The buffer bootstraps the "bottom" of the component to the output of the filter, which will improve upon the simple two divider case. This interpretation is the reason why Sallen-Key filters are often drawn with the operational amplifier's non-inverting input below the inverting input, thus emphasizing the similarity between the output and ground.By choosing different passive components (
e.g. ,resistor s andcapacitor s) for , , , and , the filter can be made with low-pass, bandpass, and high-pass characteristics. To derive the expression shown above, note that the non-inverting input (i.e. , the input) matches the output. It is safe to make this assumption because the ideal operational amplifier hasnegative feedback and is connected as a unity-gain voltage follower.To apply this analysis to the specific examples below, recall that a resistor with resistance has impedance of:and a capacitor with
capacitance has impedance of:where and is afrequency of a puresine wave input. That is, a capacitor's impedance is frequency dependent and a resistor's impedance is not.Low-pass configuration
An example of the unity-gain low-pass configuration is shown below:
An
operational amplifier is used as the buffer here, although anemitter follower is also effective. This circuit is equivalent to the generic case above with:
The
transfer function for this second-order unity-gain low-pass filter is:
where the
cutoff frequency andQ factor are given by:
and
:
That is,
:
The factor determines the height and width of the peak of the
frequency response of the filter. As this parameter increases, the filter will tend to "ring" at a single resonantfrequency near (see "LC filter " for a related discussion).A designer must choose the and appropriate for his application. For example, a second-order
Butterworth filter , which has maximally flat passband frequency response, has a of . Because there are two parameters and four unknowns, the design procedure typically fixes one resistor as a ratio of the other resistor and one capacitor as a ratio of the other capacitor. One possibility is to set the ratio between and as and the ratio between and as . So,:
:
:
:
Therefore, the and expressions are
:
and
:
For example, the above circuit has an of and a of . The
transfer function is given by:
and, after substitution, this expression is equal to
:
which shows how every combination comes with some combination to provide the same and for the low-pass filter. A similar design approach is used for the other filters below.
High-pass configuration
A second-order unity-gain high-pass filter with of and of is shown below:
A second-order unity-gain high-pass filter has the transfer function
where cutoff frequency and factor are discussed above in the low-pass filter discussion. The circuit above implements this transfer function by the equations
(as before), and
Follow an approach similar to the one used to design the low-pass filter above.
Band-pass configuration
An example of the band-pass configuration is shown below:
An
operational amplifier is used here as a buffer with gain, which affects the filter's Q. Although anemitter follower might be effective, the components would need different values to have the same Q as an emitter follower has no gain.The peak frequency is given by:
:
The voltage divider in the negative feedback loop controls the gain. The "inner gain" "G" is given by
:
while the amplifier gain at the peak frequency is given by:
:
It can be seen that "G" must be kept below 3 or else the filter will oscillate. The filter is usually optimized by selecting and .
Implementation
The calculations above assume that all components used are ideal. This means, for example, that any kind of amplifier used in the implementation of the filter has infinite input
impedance and zero output impedance at all frequencies of interest. It also means that all resistors and capacitors are exactly the values stated, with zero tolerance for error. And it also means that the filter is driven by a signal source that has zero impedance at all frequencies of interest. Most of these assumptions are invalid in any actual implementation of these circuits because these ideal components do not exist. Therefore, the response of an actual filter will only approximate the theoretical response indicated by these calculations. How close this approximation is depends on how close the components utilized approximate the ideal.ee also
*
Filter design External links
* [http://focus.ti.com/lit/an/sloa024b/sloa024b.pdf Texas Instruments Application Report: Analysis of the Sallen–Key Architecture]
* [http://www.analog.com/Analog_Root/static/techSupport/designTools/interactiveTools/filter/filter.html Analog Devices filter design applet] — A simple online tool for designing active filters using voltage-feedback op-amps.
* [http://www-k.ext.ti.com/SRVS/CGI-BIN/WEBCGI.EXE/,/?St=147,E=0000000000002472277,K=2597,Sxi=1,Case=obj(26717) TI active filter design source FAQ]
* [http://focus.ti.com/lit/ml/sloa088/sloa088.pdf Op Amps for Everyone — Chapter 16]
* [http://www.postreh.com/vmichal/articles/Real_properties_of_op_amp_analysis.pdf Real properties of Sallen–Key basic block]
* [http://www.changpuak.ch/electronics/calc_08.html Online Calculation Tool for Sallen–Key Lowpass / Highpass Filters]
* [http://www.tedpavlic.com/teaching/osu/ece327/lab7_proj/lab7_proj_procedure.pdf ECE 327: Procedures for Output Filtering Lab] — Section 3 ("Smoothing Low-Pass Filter") discusses active filtering with Sallen–Key Butterworth low-pass filter.References
Wikimedia Foundation. 2010.