Sallen–Key topology

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 operational amplifier (OA) 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_out, and so

By Kirchhoff's current law (KCL) applied at the v_x node,

By combining equations (1) and (2),

Applying equation (1) and KCL at the OA's non-inverting input v₊ gives

which means that

Combining equations (2) and (3) gives

Rearranging equation (4) gives the transfer function

which typically describes a second-order LTI system.

Interpretation

If the Z 3 component were connected to ground, 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 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 operational amplifier'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 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

and a capacitor with capacitance C has impedance Z C of

where s = j ω = 2 π 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.

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 f 0 , attenuation α , and Q factor Q (i.e., damping ratio ζ ) are given by

and

So,

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).

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 s terms in the denominator).

Design choices

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 / 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 one resistor as a ratio of the other resistor and one capacitor as a ratio of the other capacitor. One possibility is to set the ratio between C 1 and C 2 as n and the ratio between R 1 and R 2 as m . So,

Therefore, the f 0 and Q expressions are

and

In practice, certain choices of component values will perform better than others due to the non-idealities of real operational amplifiers.

Example

For example, the circuit in Figure 3 has f 0 = 15.9   kHz and Q = 0.5 . The transfer function is given by

and, after substitution, this expression is equal to

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.

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

where s ′ = s ω 0 and k = R 1 R 1 + R 2 = m m + 1 .

Furthermore, for Q > 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

Fortunately, this equation is well-approximated by

for 0.25 ≤ k ≤ 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.,

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

Application: high-pass filter

A second-order unity-gain high-pass filter with f 0 = 72   Hz and Q = 0.5 is shown in Figure 4.

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

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

(as before) and

So

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

Application: bandpass filter

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

The center frequency f 0 (i.e., the frequency where the magnitude response has its peak) is given by

The Q factor Q is given by

The voltage divider in the negative feedback loop controls the "inner gain" G of the operational amplifier:

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