Sallen Key filter

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A Sallen and Key filter is a type of active filter, particularly valued for its simplicity. The circuit produces a 2-pole (12dB/octave) lowpass or highpass response using two resistors, two capacitors and a unity-gain buffer amplifier. Higher-order filters can be obtained by cascading two or more stages. This filter topology is also known as a voltage controlled voltage source (VCVS) filter. It was introduced by R.P. Sallen and E. L. Key of MIT's Lincoln Laboratory in 1955.

Although the filters depicted here have a passband gain of 1 (or 0 dB), not all Sallen and Key filters have a gain of 1 in the passband. Additional resistors can be added to the op-amp making a non-inverting amplifier with gain greater than 1. Sallen Key filters are relatively resilient to component tolerance, although obtaining high Q factor may require extreme component values or higher gain in the non-inverting amplifier.

1 Low-pass configuration
2 High-pass configuration
3 Band-pass configuration
4 See also
5 External links

[edit] Low-pass configuration

An example of the unity-gain low-pass configuration is shown below:

An operational amplifier is used as the buffer here, although an emitter follower is also effective. In general, the cutoff frequency and Q factor are given by the following equations:

$F_c = \frac{1}{2\pi\sqrt{R_1R_2C_1C_2}}$

$Q = \frac{\sqrt{R_1R_2C_1C_2}}{C_2(R_1+R_2)}$

Ratio between C_1 and C_2 are n and the ratio between R_1 and R_2 are m, thus :

R 1 = m R

R 2 = R

C 1 = n C

C 2 = C

$F_c = \frac{1}{2\pi RC\sqrt{mn}}$

$Q = \frac{\sqrt{mn}}{m+1}$

So, for example, the above circuit has an Fc of 15.9 kHz and a Q of 0.5. The transfer function is given by:

$H(s)=\frac{1}{1+C_2(R_1+R_2)s+C_1C_2R_1R_2s^2}$

$H(s)=\frac{1}{1+RC(m+1)s+mnR^2C^2s^2}$

[edit] High-pass configuration

An example of the unity gain high-pass filter with an Fc of 72Hz and Q of 0.5 is shown below.

The relevant equations are:

$F_c = \frac{1}{2\pi\sqrt{R_1R_2C_1C_2}}$

(as before), and

$Q = \frac{R_2C_x}{\sqrt{R_1R_2C_1C_2}}$

where

$C_x = \frac{C_1C_2}{C_1+C_2}$

The transfer function is given by:

$G = \frac{s^2}{s^2+2\pi(\frac{F_c}{Q})s+4\pi^2(F_c^2)}$

[edit] Band-pass configuration

An example of the band-pass configuration is shown below:

An operational amplifier is used here as a buffer with gain, which affects the filter's Q. Although an emitter follower might be effective, the components would need different values to have the same Q as an emitter follower has no gain.

The peak frequency is given by:

$F_c=\frac{1}{2\pi}\sqrt{\frac{R_f+R_1}{C_1C_2R_1R_2R_f}}$

The voltage divider in the positive feedback loop controls the gain. The "inner gain" G is given by

$G=1+\frac{R_b}{R_a}$

while the amplifier gain at the peak frequency is given by:

$A=\frac{G}{3-G}$

It can be seen that G must be kept below 3 or else the filter will oscillate. The filter is usually optimized by selecting $R 2 = 2 R 1$ and $C 1 = C 2$ .