Editing Sallen–Key topology

{{short description|Electronic filter topology}}

The '''Sallen–Key topology''' is an [[electronic filter topology]] used to implement [[Low-pass filter#Continuous-time realization|second-order]] [[active filter]]s that is particularly valued for its simplicity.<ref name="EE315A_Notes">[https://ccnet.stanford.edu/cgi-bin/course.cgi?cc=ee315a&action=handout_download&handout_id=ID126954294624704 "EE315A Course Notes - Chapter 2"-B. Murmann] {{webarchive|url=https://web.archive.org/web/20100716003151/https://ccnet.stanford.edu/cgi-bin/course.cgi?cc=ee315a&action=handout_download&handout_id=ID126954294624704 |date=2010-07-16 }}</ref> It is a [[degeneracy (mathematics)|degenerate]] form of a '''voltage-controlled voltage-source''' ('''VCVS''') '''filter topology'''.  It was introduced by [[R. P. Sallen]] and [[E. L. Key]] of [[MIT]] [[Lincoln Laboratory]] in 1955.<ref name="SallenKey">{{cite journal|title=A Practical Method of Designing RC Active Filters|journal=IRE Transactions on Circuit Theory|date=March 1955|first=R. P.|last=Sallen|author2=E. L. Key |volume=2|issue=1|pages=74–85|doi=10.1109/tct.1955.6500159|s2cid=51640910}}</ref>

==Explanation of operation==
A VCVS filter uses a voltage amplifier with practically infinite [[input impedance]] and zero [[output impedance]] to implement a [[Pole (complex analysis)|2-pole]] [[low-pass]], [[high-pass]], [[bandpass]], [[band-stop filter|bandstop]], or [[allpass]] [[frequency response|response]]. The VCVS filter allows high [[Q factor]] and [[passband]] gain without the use of [[inductor]]s. A VCVS filter also has the advantage of independence: VCVS filters can be cascaded without the stages affecting each others tuning. A Sallen–Key filter is a variation on a VCVS filter that uses a [[unity gain]] amplifier (i.e., a [[buffer amplifier]]).

==History and implementation==
In 1955, Sallen and Key used [[vacuum tube]] [[cathode follower]] amplifiers; the cathode follower is a reasonable approximation to an amplifier with unity voltage gain. Modern analog filter implementations may use [[operational amplifier]]s (also called ''op amps''). Because of its high input impedance and easily selectable gain, an operational amplifier in a conventional [[Operational amplifier applications#Non-inverting amplifier|non-inverting configuration]] is often used in VCVS implementations.{{Citation needed|date=January 2009}} Implementations of Sallen–Key filters often use an op amp configured as a [[Operational amplifier applications#Voltage follower|voltage follower]]; however, [[common collector|emitter]] or [[common drain|source]] followers are other common choices for the buffer amplifier.

== Sensitivity to component tolerances==
VCVS filters are relatively resilient to component [[tolerance (engineering)|tolerance]], but obtaining high Q factor may require extreme component value spread or high amplifier gain.<ref name="EE315A_Notes" /> Higher-order filters can be obtained by cascading two or more stages.

==Generic Sallen–Key topology{{anchor|Generic Sallen–Key topology|Generic Sallen-Key topology}}==
[[File:Sallen-Key Generic Circuit.svg|thumb|upright=1.4|Figure&nbsp;1: The generic Sallen–Key filter topology]]
The generic unity-gain Sallen–Key filter topology implemented with a unity-gain operational amplifier is shown in Figure&nbsp;1. The following analysis is based on the assumption that the operational amplifier is ideal.

Because the op amp is in a [[negative feedback|negative-feedback]] configuration, its <math>v_+</math> and <math>v_-</math> inputs must match (i.e., <math>v_+ = v_-</math>). However, the inverting input <math>v_-</math> is connected directly to the output <math>v_\text{out}</math>, and so

{{NumBlk|:|<math>v_+ = v_- = v_\text{out}.</math>|1}}

By [[Kirchhoff's circuit laws|Kirchhoff's current law]] (KCL) applied at the <math>v_x</math> node,

{{NumBlk|:|<math>\frac{v_\text{in} - v_x}{Z_1} = \frac{v_x-v_\text{out}}{Z_3} + \frac{v_x - v_-}{Z_2}.</math>|2}}

By combining equations&nbsp;(1) and&nbsp;(2),

:<math>\frac{v_\text{in} - v_x}{Z_1} = \frac{v_x - v_\text{out}}{Z_3} + \frac{v_x - v_\text{out}}{Z_2}.</math>

Applying equation&nbsp;(1) and KCL at the op amp's non-inverting input <math>v_+</math> gives

:<math>\frac{v_x - v_\text{out}}{Z_2} = \frac{v_\text{out}}{Z_4},</math>

which means that

{{NumBlk|:|<math>v_x = v_\text{out} \left( \frac{Z_2}{Z_4} + 1 \right).</math>|3}}

Combining equations&nbsp;(2) and&nbsp;(3) gives

{{NumBlk|:|<math>\frac{v_\text{in} - v_\text{out} \left( \frac{Z_2}{Z_4} + 1 \right)}{Z_1} = \frac{v_\text{out} \left( \frac{Z_2}{Z_4} + 1 \right) - v_\text{out}}{Z_3} + \frac{v_\text{out} \left( \frac{Z_2}{Z_4} + 1 \right) - v_\text{out}}{Z_2}.</math>|4}}

Rearranging equation&nbsp;(4) gives the [[transfer function]]

{{NumBlk|:|<math>\frac{v_\text{out}}{v_\text{in}} = \frac{Z_3 Z_4}{Z_1 Z_2 + Z_3(Z_1 + Z_2) + Z_3 Z_4},</math>|5}}

which typically describes a second-order [[LTI system theory|linear time-invariant (LTI) system]].

If the <math>Z_3</math> component were connected to ground instead of to <math>v_\text{out}</math>, the filter would be a [[voltage divider]] composed of the <math>Z_1</math> and <math>Z_3</math> components cascaded with another voltage divider composed of the <math>Z_2</math> and <math>Z_4</math> components. The buffer amplifier [[Bootstrapping (electronics)|bootstraps]] the "bottom" of the <math>Z_3</math> 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 op amp's non-inverting input below the inverting input, thus emphasizing the similarity between the output and ground.

===Branch impedances===

By choosing different [[Passivity (engineering)|passive components]] (e.g., [[resistor]]s and [[capacitor]]s) for <math>Z_1</math>, <math>Z_2</math>, <math>Z_4</math>, and <math>Z_3</math>, the filter can be made with [[#Application:_low-pass_filter|low-pass]], [[#Application:_bandpass_filter|bandpass]], and [[#Application:_high-pass_filter|high-pass]] characteristics. In the examples below, recall that a resistor with [[electrical resistance|resistance]] <math>R</math> has [[electrical impedance|impedance]] <math>Z_R</math> of
:<math>Z_R = R,</math>
and a capacitor with [[capacitance]] <math>C</math> has impedance <math>Z_C</math> of
:<math>Z_C = \frac{1}{s C},</math>
where <math>s = j \omega = 2 \pi j f</math> (here <math>j</math> denotes the [[imaginary unit]]) is the [[complex number|complex]] [[angular frequency]], and <math>f</math> is the [[frequency]] of a pure [[sine-wave]] input. That is, a capacitor's impedance is frequency-dependent and a resistor's impedance is not.

==Application: low-pass filter{{anchor|Low-pass configuration}}==
[[File:Sallen-Key Lowpass General.svg|thumb|upright=1.2|Figure&nbsp;2: A unity-gain low-pass filter implemented with a Sallen–Key topology]]
An example of a unity-gain low-pass configuration is shown in Figure&nbsp;2.
An operational amplifier is used as the buffer here, although an [[emitter follower]] is also effective. This circuit is equivalent to the generic case above with

:<math>Z_1 = R_1, \quad Z_2 = R_2, \quad Z_3 = \frac{1}{s C_1}, \quad Z_4 = \frac{1}{s C_2}.</math>

The transfer function for this second-order unity-gain low-pass filter is

:<math>H(s) = \frac{\omega_0^2}{s^2 + 2 \alpha s + \omega_0^2},</math>

where the undamped [[natural frequency]] <math>f_0</math>, [[attenuation coefficient|attenuation]] <math>\alpha</math>, Q factor <math>Q</math>, and [[damping ratio]] <math>\zeta</math>, are given by

:<math> \omega_0 =  2 \pi f_0 = \frac{1}{ \sqrt{R_1 R_2 C_1 C_2} } </math>

and

:<math> 2 \alpha = 2 \zeta \omega_0 = \frac{\omega_0}{Q} = \frac{1}{C_1} \left( \frac{1}{R_1} + \frac{1}{R_2} \right)  
= \frac{1}{C_1} \left( \frac{R_1 + R_2}{R_1 R_2} \right).</math>

So,

:<math> Q = \frac{\omega_0}{2 \alpha} = \frac{\sqrt{R_1 R_2 C_1 C_2}}{C_2 \left( R_1 + R_2 \right)}.</math>

The <math>Q</math> 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 [[resonant frequency]] near <math>f_0</math> (see "[[LC filter]]" for a related discussion).

===Poles and zeros===

This transfer function has no (finite) zeros and two [[pole–zero plot|poles]] located in the complex [[s plane|''s''-plane]]:

:<math> s = -\alpha \pm \sqrt{\alpha^2 - \omega_0^2}.</math>

There are two zeros at infinity (the transfer function goes to zero for each of the <math>s</math> terms in the denominator).

=== Design choices===
A [[filter design|designer]] must choose the <math>Q</math> and <math>f_0</math> appropriate for their application.
The <math>Q</math> value is critical in determining the eventual shape.
For example, a second-order [[Butterworth filter]], which has maximally flat passband frequency response, has a <math>Q</math> of <math>1/\sqrt{2}</math>.
By comparison, a value of <math>Q = 1/2</math> corresponds to the series cascade of two identical simple low-pass filters.

Because there are 2 parameters and 4 unknowns, the design procedure typically fixes the ratio between both resistors as well as that between the capacitors. One possibility is to set the ratio between <math>C_1</math> and <math>C_2</math> as <math>n</math> versus <math>1/n</math> and the ratio between <math>R_1</math> and <math>R_2</math> as <math>m</math> versus <math>1/m</math>. So,

:<math>
\begin{align}
 R_1 &= mR, \\
 R_2 &= R/m, \\
 C_1 &= nC, \\
 C_2 &= C/n.
\end{align}
</math>

As a result, the <math>f_0</math> and <math>Q</math> expressions are reduced to

:<math> \omega_0 = 2 \pi f_0 = \frac{1}{RC}</math>

and

:<math> Q = \frac{mn}{m^2 + 1}. </math>

[[File:Sallen-Key Lowpass Example.svg|thumb|upright=1.2|Figure&nbsp;3: A low-pass filter, which is implemented with a Sallen–Key topology, with ''f''<sub>0</sub>&nbsp;=&nbsp;15.9&nbsp;kHz and ''Q''&nbsp;=&nbsp;0.5]]

Starting with a more or less arbitrary choice for e.g. <math>C</math> and <math>n</math>, the appropriate values for <math>R</math> and <math>m</math> can be calculated in favor of the desired <math>f_0</math> and <math>Q</math>. In practice, certain choices of component values will perform better than others due to the non-idealities of real operational amplifiers.<ref>[http://www.ti.com/lit/an/slyt306/slyt306.pdf Stop-band limitations of the Sallen–Key low-pass filter].</ref> As an example, high resistor values will increase the circuit's noise production, whilst contributing to the DC offset voltage on the output of op amps equipped with bipolar input transistors.

===Example===
For example, the circuit in Figure&nbsp;3 has <math>f_0 = 15.9~\text{kHz}</math> and <math>Q = 0.5</math>. The transfer function is given by

:<math>H(s) = \frac{1}{1 + \underbrace{C_2(R_1 + R_2)}_{\frac{2 \zeta}{\omega_0} = \frac{1}{\omega_0 Q}}s + \underbrace{C_1 C_2 R_1 R_2}_{\frac{1}{\omega_0^2}}s^2},</math>

and, after the substitution, this expression is equal to

:<math>H(s) = \frac{1}{1 + \underbrace{\frac{RC(m+1/m)}{n}}_{\frac{2 \zeta}{\omega_0} = \frac{1}{\omega_0 Q}}s + \underbrace{R^2 C^2}_{\frac{1}{{\omega_0}^2}}s^2},</math>

which shows how every <math>(R,C)</math> combination comes with some <math>(m,n)</math> combination to provide the same <math>f_0</math> and <math>Q</math> for the low-pass filter. A similar design approach is used for the other filters below.

=== Input impedance ===
The input impedance of the second-order unity-gain Sallen–Key low-pass filter is also of interest to designers. It is given by Eq.&nbsp;(3) in Cartwright and Kaminsky<ref name="Cartwright and Kaminsky">{{cite journal|last=Cartwright|first=K. V.|author2=E. J. Kaminsky |title=Finding the minimum input impedance of a second-order unity-gain Sallen-Key low-pass filter without calculus|journal=Lat. Am. J. Phys. Educ.|year=2013|volume=7|issue=4|pages=525–535|url=http://www.lajpe.org/dec13/3-LAJPE_828_Kenneth_Cartwrigth.pdf}}</ref> as

:<math>Z(s) = R_1\frac{s'^2 + s'/Q + 1}{s'^2 + s'k/Q},</math>

where <math>s' = \frac{s}{\omega_0}</math> and <math>k = \frac{R_1}{R_1 + R_2} = \frac{m}{m+1/m}</math>.

Furthermore, for <math>Q>\sqrt{\frac{1 - k^2}{2}}</math>, there is a minimal value of the magnitude of the impedance, given by Eq.&nbsp;(16) of Cartwright and Kaminsky,<ref name="Cartwright and Kaminsky" /> which states that

:<math>|Z(s)|_\text{min} = R_1\sqrt{1 - \frac{(2Q^2 + k^2 - 1)^2}{2Q^4 + k^2(2Q^2 + k^2 - 1)^2 + 2Q^2\sqrt{Q^4 + k^2(2Q^2 + k^2 - 1)}}}.</math>

Fortunately, this equation is well-approximated by<ref name="Cartwright and Kaminsky" />

:<math>|Z(s)|_\text{min} \approx R_1\sqrt{\frac{1}{Q^2 + k^2 + 0.34}}</math>

for <math>0.25\leq k \leq 0.75</math>. For <math>k</math> values outside of this range, the 0.34 constant has to be modified for minimal error.

Also, the frequency at which the minimal impedance magnitude occurs is given by Eq.&nbsp;(15) of Cartwright and Kaminsky,<ref name="Cartwright and Kaminsky" /> i.e.,

:<math>\omega_\text{min} = \omega_0\sqrt{\frac{Q^2 + \sqrt{Q^4 + k^2(2Q^2 + k^2 - 1)}}{2Q^2 + k^2 - 1}}.</math>

This equation can also be well approximated using Eq.&nbsp;(20) of Cartwright and Kaminsky,<ref name="Cartwright and Kaminsky" /> which states that

:<math>\omega_\text{min} \approx \omega_0\sqrt{\frac{2Q^2}{2Q^2 + k^2 - 1}}.</math>

==Application: high-pass filter==
[[File:Sallen-Key Highpass Example.svg|thumb|upright=1.4|Figure&nbsp;4: A specific Sallen–Key high-pass filter with ''f''<sub>0</sub>&nbsp;=&nbsp;72&nbsp;Hz and ''Q''&nbsp;=&nbsp;0.5]]
A second-order unity-gain high-pass filter with <math>f_0 = 72~\text{Hz}</math> and <math>Q = 0.5</math> is shown in Figure&nbsp;4.

A second-order unity-gain high-pass filter has the transfer function

:<math> H(s) = \frac{s^2}{s^2 + \underbrace{2\pi\left(\frac{f_0}{Q}\right)}_{2\zeta\omega_0 = \frac{\omega_0}{Q}}s + \underbrace{(2\pi f_0)^2}_{\omega_0^2}}, </math>

where undamped natural frequency <math>f_0</math> and <math>Q</math> factor are discussed above in the [[#Low-pass configuration|low-pass filter]] discussion. The circuit above implements this transfer function by the equations

:<math> \omega_0 = 2 \pi f_0 = \frac{1}{\sqrt{R_1 R_2 C_1 C_2}}</math>

(as before) and

:<math> \frac{1}{2\zeta} = Q = \frac{\omega_0}{2\alpha} = \frac{\sqrt{R_1 R_2 C_1 C_2}}{R_1(C_1 + C_2)}.</math>

So

:<math> 2\alpha = 2\zeta\omega_0 = \frac{\omega_0}{Q} = \frac{C_1 + C_2}{R_2 C_1 C_2}. </math>

Follow an approach similar to the one used to design the low-pass filter above.

== Application: bandpass filter== 
[[File:VCVS Filter Bandpass General.svg|thumb|upright=1.2|Figure&nbsp;5: A bandpass filter realized with a VCVS topology]]
An example of a non-unity-gain bandpass filter implemented with a VCVS filter is shown in Figure&nbsp;5. Although it uses a different topology and an operational amplifier configured to provide non-unity-gain, it can be analyzed using similar methods as with the [[#Generic Sallen–Key topology|generic Sallen–Key topology]]. Its transfer function is given by

:<math>H(s) = \frac{\overbrace{\left(1 + \frac{R_\text{b}}{R_\text{a}}\right)}^{G} \frac{s}{R_1 C_1}}{s^2 +
  \underbrace{\left( \frac{1}{R_1 C_1} + \frac{1}{R_2 C_1} + \frac{1}{R_2 C_2} - \frac{R_\text{b}}{R_\text{a} R_\text{f} C_1} \right)}_{2 \zeta \omega_0 = \frac{\omega_0}{Q}} s +
  \underbrace{\frac{R_1 + R_\text{f}}{R_1 R_\text{f} R_2 C_1 C_2}}_{{\omega_0}^2 = (2\pi f_0)^2}}.</math>

The [[center frequency]] <math>f_0</math> (i.e., the frequency where the magnitude response has its ''peak'') is given by

:<math> f_0 = \frac{1}{2\pi}\sqrt{\frac{R_\text{f} + R_1}{C_1 C_2 R_1 R_2 R_\text{f}}}. </math>

The Q factor <math>Q</math> is given by

:<math> \begin{align} Q
&= \frac{\omega_0}{2 \zeta \omega_0}
= \frac{\omega_0}{\omega_0/Q}\\[10pt]
&= \frac{\sqrt{\frac{R_1 + R_\text{f}}{R_1 R_\text{f} R_2 C_1 C_2}}}{ \frac{1}{R_1 C_1} + \frac{1}{R_2 C_1} + \frac{1}{R_2 C_2} - \frac{R_\text{b}}{R_\text{a} R_\text{f} C_1} }\\[10pt]
&= \frac{\sqrt{ (R_1 + R_\text{f}) R_1 R_\text{f} R_2 C_1 C_2}}{ R_1 R_\text{f} (C_1 + C_2) + R_2 C_2 \left( R_\text{f} - \frac{R_\text{b}}{R_\text{a}} R_1 \right) }.
\end{align}</math>

The voltage divider in the negative feedback loop controls the "inner gain" <math>G</math> of the op amp:

:<math> G = 1 + \frac{R_\text{b}}{R_\text{a}}.</math>

If the inner gain <math>G</math> is too high, the filter will oscillate.

==See also==
* [[Filter design]]
* [[Electronic filter topology]]
* [[Harmonic oscillator]]
* [[Resonance]]

==References==
{{reflist}}

==External links==
* [http://focus.ti.com/lit/an/sloa024b/sloa024b.pdf Texas Instruments Application Report: Analysis of the Sallen&ndash;Key Architecture]
* [http://www.analog.com/designtools/en/filterwizard/ Analog Devices filter design tool]&nbsp;&ndash; 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]
* [https://web.archive.org/web/20060616013258/http://focus.ti.com/lit/ml/sloa088/sloa088.pdf Op Amps for Everyone&nbsp;&ndash; Chapter 16]
* [http://postreh.com/vmichal/papers/frequency_filters_with_high_attenuation_Radio2009VMJS.pdf High frequency modification of Sallen-Key filter - improving the stopband attenuation floor]
* [http://www.changpuak.ch/electronics/calc_08.php Online Calculation Tool for Sallen&ndash;Key Low-pass/High-pass Filters]
* [http://sim.okawa-denshi.jp/en/Fkeisan.htm Online Calculation Tool for Filter Design and Analysis]
* [http://www.tedpavlic.com/teaching/osu/ece327/lab7_proj/lab7_proj_procedure.pdf ECE 327: Procedures for Output Filtering Lab]&nbsp;&ndash; Section 3 ("Smoothing Low-Pass Filter") discusses active filtering with Sallen&ndash;Key Butterworth low-pass filter.
* [https://www.youtube.com/watch?v=DEV7l66D6Ys Filtering 101: Multi Pole Filters with Sallen-Key], Matt Duff of Analog Devices explains how Sallen Key circuit works

{{DEFAULTSORT:Sallen-Key Topology}}
[[Category:Linear filters]]
[[Category:Electronic filter topology]]