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 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 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. 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 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 Qs + \underbrace{C_1 C_2 R_1 R_2}_{\frac{1}{\omega_0^2s^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 Qs + \underbrace{R^2 C^2}_{\frac{1}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. (3) in Cartwright and Kaminsky 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. (16) of Cartwright and Kaminsky,