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-voltage-gain amplifier (i.e., a pure 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. 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.[1] Higher-order filters can be obtained by cascading two or more stages.

Figure 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 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 ${\displaystyle v_{+}}$ and ${\displaystyle v_{-}}$ inputs must match (i.e., ${\displaystyle v_{+}=v_{-}}$). However, the inverting input ${\displaystyle v_{-}}$ is connected directly to the output ${\displaystyle v_{\text{out}}}$, and so

${\displaystyle v_{+}=v_{-}=v_{\text{out}}.}$

(1)

By Kirchhoff's current law (KCL) applied at the ${\displaystyle v_{x}}$ node,

${\displaystyle {\frac {v_{\text{in}}-v_{x}}{Z_{1}}}={\frac {v_{x}-v_{\text{out}}}{Z_{3}}}+{\frac {v_{x}-v_{-}}{Z_{2}}}.}$

(2)

By combining equations (1) and (2),

${\displaystyle {\frac {v_{\text{in}}-v_{x}}{Z_{1}}}={\frac {v_{x}-v_{\text{out}}}{Z_{3}}}+{\frac {v_{x}-v_{\text{out}}}{Z_{2}}}.}$

Applying equation (1) and KCL at the op amp's non-inverting input ${\displaystyle v_{+}}$ gives

${\displaystyle {\frac {v_{x}-v_{\text{out}}}{Z_{2}}}={\frac {v_{\text{out}}}{Z_{4}}},}$

which means that

${\displaystyle v_{x}=v_{\text{out}}\left({\frac {Z_{2}}{Z_{4}}}+1\right).}$

(3)

Combining equations (2) and (3) gives

${\displaystyle {\frac {v_{\text{in}}-v_{\text{out}}\left({\frac {Z_{2}}{Z_{4}}}+1\right)}{Z_{1}}}={\frac {v_{\text{out}}\left({\frac {Z_{2}}{Z_{4}}}+1\right)-v_{\text{out}}}{Z_{3}}}+{\frac {v_{\text{out}}\left({\frac {Z_{2}}{Z_{4}}}+1\right)-v_{\text{out}}}{Z_{2}}}.}$

(4)

Rearranging equation (4) gives the transfer function

${\displaystyle {\frac {v_{\text{out}}}{v_{\text{in}}}}={\frac {Z_{3}Z_{4}}{Z_{1}Z_{2}+Z_{3}(Z_{1}+Z_{2})+Z_{3}Z_{4}}},}$

(5)

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

If the ${\displaystyle Z_{3}}$ component were connected to ground instead of to ${\displaystyle v_{\text{out}}}$, the filter would be a voltage divider composed of the ${\displaystyle Z_{1}}$ and ${\displaystyle Z_{3}}$ components cascaded with another voltage divider composed of the ${\displaystyle Z_{2}}$ and ${\displaystyle Z_{4}}$ components. The buffer amplifier bootstraps the "bottom" of the ${\displaystyle Z_{3}}$ 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.

Figure 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 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

${\displaystyle Z_{1}=R_{1},\quad Z_{2}=R_{2},\quad Z_{3}={\frac {1}{sC_{1}}},\quad Z_{4}={\frac {1}{sC_{2}}}.}$

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

${\displaystyle H(s)={\frac {\omega _{0}^{2}}{s^{2}+2\alpha s+\omega _{0}^{2}}},}$

where the undamped natural frequency ${\displaystyle f_{0}}$, attenuation ${\displaystyle \alpha }$, Q factor ${\displaystyle Q}$, and damping ratio ${\displaystyle \zeta }$, are given by

${\displaystyle \omega _{0}=2\pi f_{0}={\frac {1}{\sqrt {R_{1}R_{2}C_{1}C_{2}}}}}$

and

${\displaystyle 2\alpha =2\zeta \omega _{0}={\frac {\omega _{0}}{Q}}={\frac {1}{C_{1}}}\left({\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}\right)={\frac {1}{C_{1}}}\left({\frac {R_{1}+R_{2}}{R_{1}R_{2}}}\right).}$

So,

${\displaystyle Q={\frac {\omega _{0}}{2\alpha }}={\frac {\sqrt {R_{1}R_{2}C_{1}C_{2}}}{C_{2}\left(R_{1}+R_{2}\right)}}.}$

The ${\displaystyle Q}$ 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 ${\displaystyle f_{0}}$ (see "LC filter" for a related discussion).

Design choices

A designer must choose the ${\displaystyle Q}$ and ${\displaystyle f_{0}}$ appropriate for their application. The ${\displaystyle Q}$ value is critical in determining the eventual shape. For example, a second-order Butterworth filter, which has maximally flat passband frequency response, has a ${\displaystyle Q}$ of ${\displaystyle 1/{\sqrt {2}}}$. By comparison, a value of ${\displaystyle Q=1/2}$ 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 ${\displaystyle C_{1}}$ and ${\displaystyle C_{2}}$ as ${\displaystyle n}$ versus ${\displaystyle 1/n}$ and the ratio between ${\displaystyle R_{1}}$ and ${\displaystyle R_{2}}$ as ${\displaystyle m}$ versus ${\displaystyle 1/m}$. So,

${\displaystyle {\begin{aligned}R_{1}&=mR,\\R_{2}&=R/m,\\C_{1}&=nC,\\C_{2}&=C/n.\end{aligned}}}$

As a result, the ${\displaystyle f_{0}}$ and ${\displaystyle Q}$ expressions are reduced to

${\displaystyle \omega _{0}=2\pi f_{0}={\frac {1}{RC}}}$

and

${\displaystyle Q={\frac {mn}{m^{2}+1}}.}$

Figure 3: A low-pass filter, which is implemented with a Sallen–Key topology, with f₀ = 15.9 kHz and Q = 0.5

Starting with a more or less arbitrary choice for e.g. ${\displaystyle C}$ and ${\displaystyle n}$, the appropriate values for ${\displaystyle R}$ and ${\displaystyle m}$ can be calculated in favor of the desired ${\displaystyle f_{0}}$ and ${\displaystyle Q}$. In practice, certain choices of component values will perform better than others due to the non-idealities of real operational amplifiers.[3] 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.

Figure 4: A specific Sallen–Key high-pass filter with f₀ = 72 Hz and Q = 0.5

A second-order unity-gain high-pass filter with ${\displaystyle f_{0}=72~{\text{Hz}}}$ and ${\displaystyle Q=0.5}$ is shown in Figure 4.

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

${\displaystyle H(s)={\frac {s^{2}}{s^{2}+\underbrace {2\pi \left({\frac {f_{0}}{Q}}\right)} _{2\zeta \omega _{0}={\frac {\omega _{0}}{Q}}}s+\underbrace {(2\pi f_{0})^{2}} _{\omega _{0}^{2}}}},}$

where undamped natural frequency ${\displaystyle f_{0}}$ and ${\displaystyle Q}$ factor are discussed above in the low-pass filter discussion. The circuit above implements this transfer function by the equations

${\displaystyle \omega _{0}=2\pi f_{0}={\frac {1}{\sqrt {R_{1}R_{2}C_{1}C_{2}}}}}$

(as before) and

${\displaystyle {\frac {1}{2\zeta }}=Q={\frac {\omega _{0}}{2\alpha }}={\frac {\sqrt {R_{1}R_{2}C_{1}C_{2}}}{R_{1}(C_{1}+C_{2})}}.}$

So

${\displaystyle 2\alpha =2\zeta \omega _{0}={\frac {\omega _{0}}{Q}}={\frac {C_{1}+C_{2}}{R_{2}C_{1}C_{2}}}.}$

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

Figure 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 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

${\displaystyle H(s)={\frac {\overbrace {\left(1+{\frac {R_{\text{b}}}{R_{\text{a}}}}\right)} ^{G}{\frac {s}{R_{1}C_{1}}}}{s^{2}+\underbrace {\left({\frac {1}{R_{1}C_{1}}}+{\frac {1}{R_{2}C_{1}}}+{\frac {1}{R_{2}C_{2}}}-{\frac {R_{\text{b}}}{R_{\text{a}}R_{\text{f}}C_{1}}}\right)} _{2\zeta \omega _{0}={\frac {\omega _{0}}{Q}}}s+\underbrace {\frac {R_{1}+R_{\text{f}}}{R_{1}R_{\text{f}}R_{2}C_{1}C_{2}}} _{{\omega _{0}}^{2}=(2\pi f_{0})^{2}}}}.}$

The center frequency ${\displaystyle f_{0}}$ (i.e., the frequency where the magnitude response has its peak) is given by

${\displaystyle f_{0}={\frac {1}{2\pi }}{\sqrt {\frac {R_{\text{f}}+R_{1}}{C_{1}C_{2}R_{1}R_{2}R_{\text{f}}}}}.}$

The Q factor ${\displaystyle Q}$ is given by

${\displaystyle {\begin{aligned}Q&={\frac {\omega _{0}}{2\zeta \omega _{0}}}={\frac {\omega _{0}}{\omega _{0}/Q}}\\[10pt]&={\frac {\sqrt {\frac {R_{1}+R_{\text{f}}}{R_{1}R_{\text{f}}R_{2}C_{1}C_{2}}}}{{\frac {1}{R_{1}C_{1}}}+{\frac {1}{R_{2}C_{1}}}+{\frac {1}{R_{2}C_{2}}}-{\frac {R_{\text{b}}}{R_{\text{a}}R_{\text{f}}C_{1}}}}}\\[10pt]&={\frac {\sqrt {(R_{1}+R_{\text{f}})R_{1}R_{\text{f}}R_{2}C_{1}C_{2}}}{R_{1}R_{\text{f}}(C_{1}+C_{2})+R_{2}C_{2}\left(R_{\text{f}}-{\frac {R_{\text{b}}}{R_{\text{a}}}}R_{1}\right)}}.\end{aligned}}}$

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

${\displaystyle G=1+{\frac {R_{\text{b}}}{R_{\text{a}}}}.}$

If the inner gain ${\displaystyle G}$ is too high, the filter will oscillate.

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

• Texas Instruments Application Report: Analysis of the Sallen–Key Architecture
• Analog Devices filter design tool – A simple online tool for designing active filters using voltage-feedback op-amps.
• TI active filter design source FAQ
• Op Amps for Everyone – Chapter 16
• High frequency modification of Sallen-Key filter - improving the stopband attenuation floor
• Online Calculation Tool for Sallen–Key Low-pass/High-pass Filters
• Online Calculation Tool for Filter Design and Analysis
• ECE 327: Procedures for Output Filtering Lab – Section 3 ("Smoothing Low-Pass Filter") discusses active filtering with Sallen–Key Butterworth low-pass filter.
• Filtering 101: Multi Pole Filters with Sallen-Key, Matt Duff of Analog Devices explains how Sallen Key circuit works