The Sallen–Key topology is an electronic filter topology used to implement second-order active filters that is particularly valued for its simplicity. 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.
Explanation of operation
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).
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 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.
Sensitivity to component tolerances
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.
Generic Sallen–Key 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 and inputs must match (i.e., ). However, the inverting input is connected directly to the output , and so
By Kirchhoff's current law (KCL) applied at the node,
By combining equations (1) and (2),
Applying equation (1) and KCL at the op amp's non-inverting input gives
which means that
Combining equations (2) and (3) gives
Rearranging equation (4) gives the transfer function
which typically describes a second-order linear time-invariant (LTI) system.
If the component were connected to ground instead of to , the filter would be a voltage divider composed of the and components cascaded with another voltage divider composed of the and components. The buffer amplifier bootstraps the "bottom" of the 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 passive components (e.g., resistors and capacitors) for , , , and , the filter can be made with low-pass, bandpass, and high-pass characteristics. In the examples below, recall that a resistor with resistance has impedance of
and a capacitor with capacitance has impedance of
where (here denotes the imaginary unit) is the complex angular frequency, and 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
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
The transfer function for this second-order unity-gain low-pass filter is
where the undamped natural frequency , attenuation , Q factor , and damping ratio , are given by
and
So,
The 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 (see "LC filter" for a related discussion).
Poles and zeros
This transfer function has no (finite) zeros and two poles located in the complex s-plane:
There are two zeros at infinity (the transfer function goes to zero for each of the terms in the denominator).