Sallen–Key topology

[1] It is a 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.

[2] 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.

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] 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.

The following analysis is based on the assumption that the operational amplifier is ideal.

component to the output of the filter, which will improve upon the simple two-divider case.

denotes the imaginary unit) is the complex angular frequency, and

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 The transfer function for this second-order unity-gain low-pass filter is where the undamped natural frequency

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

For example, a second-order Butterworth filter, which has maximally flat passband frequency response, has a

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.

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.

The transfer function is given by and, after the substitution, this expression is equal to which shows how every

The input impedance of the second-order unity-gain Sallen–Key low-pass filter is also of interest to designers.

(16) of Cartwright and Kaminsky,[4] which states that Fortunately, this equation is well-approximated by[4] for

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.

(20) of Cartwright and Kaminsky,[4] which states that A second-order unity-gain high-pass filter with

A second-order unity-gain high-pass filter has the transfer function where undamped natural frequency

The circuit above implements this transfer function by the equations (as before) and So Follow an approach similar to the one used to design the low-pass filter above.

is given by The voltage divider in the negative feedback loop controls the "inner gain"

Figure 1: The generic Sallen–Key filter topology
Figure 2: A unity-gain low-pass filter implemented with a Sallen–Key topology
Figure 3: A low-pass filter, which is implemented with a Sallen–Key topology, with f 0 = 15.9 kHz and Q = 0.5
Figure 4: A specific Sallen–Key high-pass filter with f 0 = 72 Hz and Q = 0.5
Figure 5: A bandpass filter realized with a VCVS topology