Sallen Key Filter

Sallen-Key filter structure of a low-pass filter, positive feedback structure

Sallen-Key filter structure of a high-pass filter, positive feedback structure

A Sallen-Key filter , also known as a Sallen and Key Filter or a Voltage Controlled Voltage Source Filter (VCVS Filter), is an active electronic filter that consists of an operational amplifier and several electrical resistors and capacitors . The name is derived from the names of the two developers RP Sallen and EL Key, who developed this filter in 1955 at the Massachusetts Institute of Technology (MIT).

The advantage of the Sallen-Key filter is that an analog filter - as shown in the adjacent figures - can be implemented with a minimum of circuitry. By adapting the filter structure, both low-pass filters , high-pass filters and band-pass filters can be implemented. Since the transfer function cannot have complex zeros due to the branch structure, only so-called all - pole filters, filters without transfer zeros, can be implemented with it. Band stop filters and filters with elliptical basic elements such as Cauer filters cannot be implemented with Sallen-Key filters.

Sallen-Key filters always have a filter order of two. This means that with a low-pass or high-pass filter, the absolute frequency response experiences a change of 12 dB per octave . Higher filter orders require a series connection of several Sallen-Key filters, with which only straight filter orders can be achieved. In Sallen-Key filters, the operational amplifier is usually operated with a positive gain factor, also referred to as a positive feedback structure. Since the conjugate-complex poles of the transfer function can also be implemented with a negative gain value, Sallen-Key filters can also be implemented in a negative feedback structure, but with a higher number of components. Due to the higher number of components with an identical transfer function, Sallen-Key filters in negative feedback structures do not play an essential role. The operational amplifier is operated with negative feedback in both cases .

The filter structure is also relatively stable with regard to component tolerances, which, in addition to the simple structure, is the reason for its practical relevance. These advantages are paid for by the disadvantage that extremely high and impractical component values are necessary to achieve the highest possible quality factor Q. Sallen-Key filters are therefore preferably used in analog circuit technology where a minimum number of components with a large tolerance scheme is important and where low quality factors are acceptable.

Application examples in the form of a low pass can be found on the analog side against aliasing immediately before the analog-digital converter and as a filter against mirror spectra (anti-imaging filter) immediately after the digital-analog converter in communications systems.

Filter design

Sallen-Key structure with voltage divider for determining the filter type

Basics

For example, if a 2nd order low pass is to be dimensioned, the transfer function is:

${\ displaystyle {\ begin {aligned} {\ underline {A}} (s_ {n}) = {\ cfrac {A_ {0}} {1+ \ omega _ {g} {\ bigl [} C_ {1} (R_ {1} + R_ {2}) + (1-A_ {0}) R_ {1} C_ {2} {\ bigl]} s_ {n} + \ omega _ {g} ^ {2} R_ { 1} R_ {2} C_ {1} C_ {2} s_ {n} ^ {2}}} \ end {aligned}}}$

With the assumption and the transfer function of the filter is enormously simplified: ${\ displaystyle R_ {1} = R_ {2} = R}$ ${\ displaystyle C_ {1} = C_ {2} = C}$

${\ displaystyle {\ begin {aligned} {\ underline {A}} (s_ {n}) = {\ cfrac {A_ {0}} {1+ \ omega _ {g} RC (3-A_ {0}) s_ {n} + (\ omega _ {g} RC) ^ {2} s_ {n} ^ {2}}} \ end {aligned}}}$

If you compare this transfer function with the general form of a 2nd order low-pass filter, the DC voltage gain can be calculated using a coefficient comparison : ${\ displaystyle A_ {0}}$

${\ displaystyle {\ begin {aligned} {\ underline {A}} (s_ {n}) = {\ cfrac {A_ {0}} {1 + a_ {1} s_ {n} + b_ {1} s_ { n} ^ {2}}} \ end {aligned}}}$

So it follows:

${\ displaystyle {\ begin {aligned} a_ {1} = \ omega _ {g} RC (3-A_ {0}) \ end {aligned}}}$

${\ displaystyle {\ begin {aligned} b_ {1} = (\ omega _ {g} RC) ^ {2} \ end {aligned}}}$

If you rearrange these two coefficients, you get:

${\ displaystyle {\ begin {aligned} RC = {\ cfrac {\ sqrt {b_ {1}}} {\ omega _ {g}}} \ end {aligned}}}$

${\ displaystyle {\ begin {aligned} A_ {0} = 3 - {\ cfrac {a_ {1}} {\ omega _ {g} RC}} = 3 - {\ cfrac {a_ {1}} {\ sqrt {b_ {1}}}} = 3 - {\ cfrac {1} {Q}} \ end {aligned}}}$

The DC gain is via the voltage divider , set and does not depend on the cutoff frequency. The filter type, however, depends on the size of . ${\ displaystyle A_ {0}}$ ${\ displaystyle R_ {3}}$ ${\ displaystyle R_ {4}}$ ${\ displaystyle A_ {0}}$

Dimensioning

So-called filter tables are used to dimension filters. The following table shows an example:

Filter coefficients for 2nd order low-pass filters
	Critical	Bessel	Butterworth	3 dB Chebyshev
${\ displaystyle a_ {1}}$	${\ displaystyle 1,2872}$	${\ displaystyle 1,3617}$	${\ displaystyle 1,4142}$	${\ displaystyle 1.0650}$
${\ displaystyle b_ {1}}$	${\ displaystyle 0,4142}$	${\ displaystyle 0.6180}$	${\ displaystyle 1.0000}$	${\ displaystyle 1.9305}$

If these parameters are used in the coefficients, the following DC voltage gains are obtained:

DC voltage gain and pole quality for low-pass filters of the 2nd order
	Critical	Bessel	Butterworth	3 dB Chebyshev	undamped
${\ displaystyle Q_ {i}}$	${\ displaystyle 0.5}$	${\ displaystyle 0.58}$	${\ displaystyle 0.71}$	${\ displaystyle 1,3}$	${\ displaystyle \ infty}$
${\ displaystyle A_ {0}}$	${\ displaystyle 1,000}$	${\ displaystyle 1,268}$	${\ displaystyle 1,586}$	${\ displaystyle 2,234}$	${\ displaystyle 3,000}$

The remaining components can then be calculated by specifying, for example, and the capacity . ${\ displaystyle R_ {4}}$ ${\ displaystyle C}$

literature

Paul Horowitz, Winfield Hill: The Art Of Electronics . Cambridge University Press, 1989, ISBN 0-521-37095-7 , pp. 267-276 .
Lutz v. Wangenheim: Active Filters and Oscillators . 1st edition. Springer, 2008, ISBN 978-3-540-71737-9 .
Ulrich Tietze, Christoph Schenk, semiconductor circuit technology , Springer-Verlag 1986, 8th edition, ISBN 3-540-16720-X (with detailed tables for the filter coefficients of cascaded Sallen-Key filters up to the 10th order)

Individual evidence

^ RP Sallen and EL Key: A Practical Method of Designing RC Active Filters , IRE Transactions on Circuit Theory, Issue 1, pages 74 to 85, 1955
↑ ^a ^b ^c Schenk, Christoph., Gamm, Eberhard., Springer-Verlag GmbH: semiconductor circuit technology . 15th, revised. and exp. Edition 2016. Berlin, ISBN 978-3-662-48354-1 , pp. 806-819 .

Web links

Commons : Sallen-Key filters - Collection of images, videos and audio files

Active Filter Design Techniques (PDF; 322 kB) Application note on analog filters and how they are calculated
Analog Devices Filter Wizard web app creates Sallen-Key filter circuits

[1] RP Sallen and EL Key: A Practical Method of Designing RC Active Filters , IRE Transactions on Circuit Theory, Issue 1, pages 74 to 85, 1955

[:0-2] Schenk, Christoph., Gamm, Eberhard., Springer-Verlag GmbH: semiconductor circuit technology . 15th, revised. and exp. Edition 2016. Berlin, ISBN 978-3-662-48354-1 , pp. 806-819 .