Sallen–Key topology

A VCVS filter uses a voltage amplifier with practically infinite input impedance and zero output impedance to implement a 2-pole low-pass, high-pass, bandpass, bandstop, or allpass response. The VCVS filter allows high Q factor and passband gain without the use of inductors. 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).

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 amplifiers (also called op amps). Because of its high input impedance and easily selectable gain, an operational amplifier in a conventional 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 voltage follower; however, emitter or source followers are other common choices for the buffer amplifier.

VCVS filters are relatively resilient to component tolerance, but obtaining high Q factor may require extreme component value spread or high amplifier gain. Higher-order filters can be obtained by cascading two or more stages.

File:Sallen-Key Generic Circuit.svg

The generic unity-gain Sallen–Key filter topology implemented with a unity-gain operational amplifier is shown in Figure 1. The following analysis is based on the assumption that the operational amplifier is ideal.

Because the op amp is in a negative-feedback configuration, its $v\_+$ and $v\_-$ inputs must match (i.e., $v\_+\; =\; v\_-$). However, the inverting input $v\_-$ is connected directly to the output $v\_\backslash text\{out\}$, and so

{{NumBlk|:|$v\_+\; =\; v\_-\; =\; v\_\backslash text\{out\}.$|1}}

By Kirchhoff's current law (KCL) applied at the $v\_x$ node,

{{NumBlk|:|$\backslash frac\{v\_\backslash text\{in\}\; -\; v\_x\}\{Z\_1\}\; =\; \backslash frac\{v\_x-v\_\backslash text\{out\}\}\{Z\_3\}\; +\; \backslash frac\{v\_x\; -\; v\_-\}\{Z\_2\}.$|2}}

By combining equations (1) and (2),

:$\backslash frac\{v\_\backslash text\{in\}\; -\; v\_x\}\{Z\_1\}\; =\; \backslash frac\{v\_x\; -\; v\_\backslash text\{out\}\}\{Z\_3\}\; +\; \backslash frac\{v\_x\; -\; v\_\backslash text\{out\}\}\{Z\_2\}.$

Applying equation (1) and KCL at the op amp's non-inverting input $v\_+$ gives

:$\backslash frac\{v\_x\; -\; v\_\backslash text\{out\}\}\{Z\_2\}\; =\; \backslash frac\{v\_\backslash text\{out\}\}\{Z\_4\},$

which means that

{{NumBlk|:|$v\_x\; =\; v\_\backslash text\{out\}\; \backslash left(\; \backslash frac\{Z\_2\}\{Z\_4\}\; +\; 1\; \backslash right).$|3}}

Combining equations (2) and (3) gives

{{NumBlk|:|$\backslash frac\{v\_\backslash text\{in\}\; -\; v\_\backslash text\{out\}\; \backslash left(\; \backslash frac\{Z\_2\}\{Z\_4\}\; +\; 1\; \backslash right)\}\{Z\_1\}\; =\; \backslash frac\{v\_\backslash text\{out\}\; \backslash left(\; \backslash frac\{Z\_2\}\{Z\_4\}\; +\; 1\; \backslash right)\; -\; v\_\backslash text\{out\}\}\{Z\_3\}\; +\; \backslash frac\{v\_\backslash text\{out\}\; \backslash left(\; \backslash frac\{Z\_2\}\{Z\_4\}\; +\; 1\; \backslash right)\; -\; v\_\backslash text\{out\}\}\{Z\_2\}.$|4}}

Rearranging equation (4) gives the transfer function

{{NumBlk|:|$\backslash frac\{v\_\backslash text\{out\}\}\{v\_\backslash text\{in\}\}\; =\; \backslash frac\{Z\_3\; Z\_4\}\{Z\_1\; Z\_2\; +\; Z\_3(Z\_1\; +\; Z\_2)\; +\; Z\_3\; Z\_4\},$|5}}

which typically describes a second-order linear time-invariant (LTI) system.

If the $Z\_3$ component were connected to ground instead of to $v\_\backslash text\{out\}$, the filter would be a voltage divider composed of the $Z\_1$ and $Z\_3$ components cascaded with another voltage divider composed of the $Z\_2$ and $Z\_4$ components. The buffer amplifier bootstraps the "bottom" of the $Z\_3$ 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.

By choosing different passive components (e.g., resistors and capacitors) for $Z\_1$, $Z\_2$, $Z\_4$, and $Z\_3$, the filter can be made with low-pass, bandpass, and high-pass characteristics. In the examples below, recall that a resistor with resistance $R$ has impedance $Z\_R$ of

:$Z\_R\; =\; R,$

and a capacitor with capacitance $C$ has impedance $Z\_C$ of

:$Z\_C\; =\; \backslash frac\{1\}\{s\; C\},$

where $s\; =\; j\; \backslash omega\; =\; 2\; \backslash pi\; j\; f$ (here $j$ denotes the imaginary unit) is the complex angular frequency, and $f$ 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.

File:Sallen-Key Lowpass General.svg

An example of a unity-gain low-pass configuration is shown in Figure 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

:$Z\_1\; =\; R\_1,\; \backslash quad\; Z\_2\; =\; R\_2,\; \backslash quad\; Z\_3\; =\; \backslash frac\{1\}\{s\; C\_1\},\; \backslash quad\; Z\_4\; =\; \backslash frac\{1\}\{s\; C\_2\}.$

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

:$H(s)\; =\; \backslash frac\{\backslash omega\_0^2\}\{s^2\; +\; 2\; \backslash alpha\; s\; +\; \backslash omega\_0^2\},$

where the undamped natural frequency $f\_0$, attenuation $\backslash alpha$, Q factor $Q$, and damping ratio $\backslash zeta$, are given by

:$\backslash omega\_0\; =\; 2\; \backslash pi\; f\_0\; =\; \backslash frac\{1\}\{\; \backslash sqrt\{R\_1\; R\_2\; C\_1\; C\_2\}\; \}$

and

:$2\; \backslash alpha\; =\; 2\; \backslash zeta\; \backslash omega\_0\; =\; \backslash frac\{\backslash omega\_0\}\{Q\}\; =\; \backslash frac\{1\}\{C\_1\}\; \backslash left(\; \backslash frac\{1\}\{R\_1\}\; +\; \backslash frac\{1\}\{R\_2\}\; \backslash right)$

= \frac{1}{C_1} \left( \frac{R_1 + R_2}{R_1 R_2} \right).

So,

:$Q\; =\; \backslash frac\{\backslash omega\_0\}\{2\; \backslash alpha\}\; =\; \backslash frac\{\backslash sqrt\{R\_1\; R\_2\; C\_1\; C\_2\}\}\{C\_2\; \backslash left(\; R\_1\; +\; R\_2\; \backslash right)\}.$

The $Q$ 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 $f\_0$ (see "LC filter" for a related discussion).

This transfer function has no (finite) zeros and two poles located in the complex s-plane:

:$s\; =\; -\backslash alpha\; \backslash pm\; \backslash sqrt\{\backslash alpha^2\; -\; \backslash omega\_0^2\}.$

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

A designer must choose the $Q$ and $f\_0$ appropriate for their application.

The $Q$ value is critical in determining the eventual shape.

For example, a second-order Butterworth filter, which has maximally flat passband frequency response, has a $Q$ of $1/\backslash sqrt\{2\}$.

By comparison, a value of $Q\; =\; 1/2$ 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 $C\_1$ and $C\_2$ as $n$ versus $1/n$ and the ratio between $R\_1$ and $R\_2$ as $m$ versus $1/m$. So,

:$$

\begin{align}

R_1 &= mR, \\

R_2 &= R/m, \\

C_1 &= nC, \\

C_2 &= C/n.

\end{align}

As a result, the $f\_0$ and $Q$ expressions are reduced to

:$\backslash omega\_0\; =\; 2\; \backslash pi\; f\_0\; =\; \backslash frac\{1\}\{RC\}$

and

:$Q\; =\; \backslash frac\{mn\}\{m^2\; +\; 1\}.$

File:Sallen-Key Lowpass Example.svg

Starting with a more or less arbitrary choice for e.g. $C$ and $n$, the appropriate values for $R$ and $m$ can be calculated in favor of the desired $f\_0$ and $Q$. In practice, certain choices of component values will perform better than others due to the non-idealities of real operational amplifiers.[http://www.ti.com/lit/an/slyt306/slyt306.pdf Stop-band limitations of the Sallen–Key low-pass filter]. 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.

For example, the circuit in Figure 3 has $f\_0\; =\; 15.9~\backslash text\{kHz\}$ and $Q\; =\; 0.5$. The transfer function is given by

:$H(s)\; =\; \backslash frac\{1\}\{1\; +\; \backslash underbrace\{C\_2(R\_1\; +\; R\_2)\}\_\{\backslash frac\{2\; \backslash zeta\}\{\backslash omega\_0\}\; =\; \backslash frac\{1\}\{\backslash omega\_0\; Q\}\}s\; +\; \backslash underbrace\{C\_1\; C\_2\; R\_1\; R\_2\}\_\{\backslash frac\{1\}\{\backslash omega\_0^2\}\}s^2\},$

and, after the substitution, this expression is equal to

:$H(s)\; =\; \backslash frac\{1\}\{1\; +\; \backslash underbrace\{\backslash frac\{RC(m+1/m)\}\{n\}\}\_\{\backslash frac\{2\; \backslash zeta\}\{\backslash omega\_0\}\; =\; \backslash frac\{1\}\{\backslash omega\_0\; Q\}\}s\; +\; \backslash underbrace\{R^2\; C^2\}\_\{\backslash frac\{1\}\{\{\backslash omega\_0\}^2\}\}s^2\},$

which shows how every $(R,C)$ combination comes with some $(m,n)$ combination to provide the same $f\_0$ and $Q$ for the low-pass filter. A similar design approach is used for the other filters below.

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. (3) in 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}} as

:$Z(s)\; =\; R\_1\backslash frac\{s\text{'}^2\; +\; s\text{'}/Q\; +\; 1\}\{s\text{'}^2\; +\; s\text{'}k/Q\},$

where $s\text{'}\; =\; \backslash frac\{s\}\{\backslash omega\_0\}$ and $k\; =\; \backslash frac\{R\_1\}\{R\_1\; +\; R\_2\}\; =\; \backslash frac\{m\}\{m+1/m\}$.

Furthermore, for $Q>\backslash sqrt\{\backslash frac\{1\; -\; k^2\}\{2\}\}$, there is a minimal value of the magnitude of the impedance, given by Eq. (16) of Cartwright and Kaminsky, which states that

:$|Z(s)|\_\backslash text\{min\}\; =\; R\_1\backslash sqrt\{1\; -\; \backslash frac\{(2Q^2\; +\; k^2\; -\; 1)^2\}\{2Q^4\; +\; k^2(2Q^2\; +\; k^2\; -\; 1)^2\; +\; 2Q^2\backslash sqrt\{Q^4\; +\; k^2(2Q^2\; +\; k^2\; -\; 1)\}\}\}.$

Fortunately, this equation is well-approximated by

:$|Z(s)|\_\backslash text\{min\}\; \backslash approx\; R\_1\backslash sqrt\{\backslash frac\{1\}\{Q^2\; +\; k^2\; +\; 0.34\}\}$

for $0.25\backslash leq\; k\; \backslash leq\; 0.75$. For $k$ 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. (15) of Cartwright and Kaminsky, i.e.,

:$\backslash omega\_\backslash text\{min\}\; =\; \backslash omega\_0\backslash sqrt\{\backslash frac\{Q^2\; +\; \backslash sqrt\{Q^4\; +\; k^2(2Q^2\; +\; k^2\; -\; 1)\}\}\{2Q^2\; +\; k^2\; -\; 1\}\}.$

This equation can also be well approximated using Eq. (20) of Cartwright and Kaminsky, which states that

:$\backslash omega\_\backslash text\{min\}\; \backslash approx\; \backslash omega\_0\backslash sqrt\{\backslash frac\{2Q^2\}\{2Q^2\; +\; k^2\; -\; 1\}\}.$

File:Sallen-Key Highpass Example.svg

A second-order unity-gain high-pass filter with $f\_0\; =\; 72~\backslash text\{Hz\}$ and $Q\; =\; 0.5$ is shown in Figure 4.

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

:$H(s)\; =\; \backslash frac\{s^2\}\{s^2\; +\; \backslash underbrace\{2\backslash pi\backslash left(\backslash frac\{f\_0\}\{Q\}\backslash right)\}\_\{2\backslash zeta\backslash omega\_0\; =\; \backslash frac\{\backslash omega\_0\}\{Q\}\}s\; +\; \backslash underbrace\{(2\backslash pi\; f\_0)^2\}\_\{\backslash omega\_0^2\}\},$

where undamped natural frequency $f\_0$ and $Q$ factor are discussed above in the low-pass filter discussion. The circuit above implements this transfer function by the equations

:$\backslash omega\_0\; =\; 2\; \backslash pi\; f\_0\; =\; \backslash frac\{1\}\{\backslash sqrt\{R\_1\; R\_2\; C\_1\; C\_2\}\}$

(as before) and

:$\backslash frac\{1\}\{2\backslash zeta\}\; =\; Q\; =\; \backslash frac\{\backslash omega\_0\}\{2\backslash alpha\}\; =\; \backslash frac\{\backslash sqrt\{R\_1\; R\_2\; C\_1\; C\_2\}\}\{R\_1(C\_1\; +\; C\_2)\}.$

So

:$2\backslash alpha\; =\; 2\backslash zeta\backslash omega\_0\; =\; \backslash frac\{\backslash omega\_0\}\{Q\}\; =\; \backslash frac\{C\_1\; +\; C\_2\}\{R\_2\; C\_1\; C\_2\}.$

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

== Application: bandpass filter==

File:VCVS Filter Bandpass General.svg

An example of a non-unity-gain bandpass filter implemented with a VCVS filter is shown in Figure 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. Its transfer function is given by

:$H(s)\; =\; \backslash frac\{\backslash overbrace\{\backslash left(1\; +\; \backslash frac\{R\_\backslash text\{b\}\}\{R\_\backslash text\{a\}\}\backslash right)\}^\{G\}\; \backslash frac\{s\}\{R\_1\; C\_1\}\}\{s^2\; +$

\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}}.

The center frequency $f\_0$ (i.e., the frequency where the magnitude response has its peak) is given by

:$f\_0\; =\; \backslash frac\{1\}\{2\backslash pi\}\backslash sqrt\{\backslash frac\{R\_\backslash text\{f\}\; +\; R\_1\}\{C\_1\; C\_2\; R\_1\; R\_2\; R\_\backslash text\{f\}\}\}.$

The Q factor $Q$ is given by

:$\backslash begin\{align\}\; Q$

&= \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}

The voltage divider in the negative feedback loop controls the "inner gain" $G$ of the op amp:

:$G\; =\; 1\; +\; \backslash frac\{R\_\backslash text\{b\}\}\{R\_\backslash text\{a\}\}.$

If the inner gain $G$ is too high, the filter will oscillate.

• Filter design
• Electronic filter topology
• Harmonic oscillator
• Resonance

{{reflist}}

• [http://focus.ti.com/lit/an/sloa024b/sloa024b.pdf Texas Instruments Application Report: Analysis of the Sallen–Key Architecture]
• [http://www.analog.com/designtools/en/filterwizard/ Analog Devices filter design tool] – 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 – 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–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] – Section 3 ("Smoothing Low-Pass Filter") discusses active filtering with Sallen–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