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 (usually) 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 Lincoln Laboratory in 1955^[1].

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. Non-unity-gain buffers can also be used (e.g., by adding additional resistors to the operational amplifier to change the feedback gain, as in 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 buffer gain).

1 Generic configuration
2 Low-pass configuration
3 High-pass configuration
4 Band-pass configuration
5 Implementation
6 See also
7 External links
8 References

[edit] Generic configuration

A generic unity-gain Sallen-and-Key filter, implemented with a unity gain operational amplifier, is shown below with an analysis that uses ideal operational amplifier theory:

If the $Z_4\,$ component was connected to ground, the filter would be a voltage divider composed of the $Z_1\,$ and $Z_4\,$ components cascaded with another voltage divider composed of the $Z_2\,$ and $Z_3\,$ components. The buffer bootstraps the "bottom" of the $Z_4\,$ 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.

By choosing different passive components (e.g., resistors and capacitors) for $Z_1\,$ , $Z_2\,$ , $Z_3\,$ , and $Z_4\,$ , the filter can be made with low-pass, bandpass, and high-pass characteristics. To derive the $v_\text{out}/v_\text{in}\,$ expression shown above, note that the non-inverting input (i.e., the $+\,$ input) matches the output. It is safe to make this assumption because the ideal operational amplifier has negative feedback and is connected as a unity-gain voltage follower.

To apply this analysis to the specific examples below, recall that a resistor with resistance $R\,$ has impedance $Z_R\,$ of

$Z_R = R\,,$

and a capacitor with capacitance $C\,$ has impedance $Z_C\,$ of

$Z_C = \frac{1}{s C}\,,$

where $s = j \omega = \left(\sqrt{-1}\right) 2 \pi f\,$ and $f\,$ is a frequency of a pure sine wave input. That is, a capacitor's impedance is frequency dependent and a resistor's impedance is not.

[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. This circuit is equivalent to the generic case above with

$Z_1 = R_1, \quad Z_2 = R_2, \quad Z_3 = \frac{1}{s C_2}, \quad \text{and} \quad Z_4 = \frac{1}{s C_1}.\,$

The transfer function for this second-order unity-gain low-pass filter is

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

where the cutoff frequency $F_c\,$ and Q factor $Q\,$ are given by

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

and

$\frac{1}{Q} = \frac{\sqrt{R_1R_2C_1C_2}}{C_1} \left( \frac{1}{R_1} + \frac{1}{R_2} \right).\,$

That is,

$Q = \frac{ \sqrt{ R_1 R_2 C_1 C_2 } }{ C_2 \left( R_1 + R_2 \right) }.\,$

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 $F_c\,$ (see "LC filter" for a related discussion).

A designer must choose the $Q\,$ and $F_c\,$ appropriate for his application. For example, a second-order Butterworth filter, which has maximally flat passband frequency response, has a $Q\,$ of $1/\sqrt{2}\,$ . Because there are two parameters and four 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,

$R_1=mR,\,$

$R_2=R,\,$

$C_1=nC,\,$

$C_2=C.\,$

Therefore, the $F_c\,$ and $Q\,$ expressions are

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

and

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

For example, the above circuit has an $F_c\,$ of $15.9\,\text{kHz}\,$ and a $Q\,$ of $0.5\,$ . The transfer function is given by

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

and, after substitution, this expression is equal to

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

which shows how every $(R,C)\,$ combination comes with some $(m,n)\,$ combination to provide the same $F c$ and $Q$ for the low-pass filter. A similar design approach is used for the other filters below.

[edit] High-pass configuration

A second-order unity-gain high-pass filter with $F_c\,$ of $72\,\text{Hz}\,$ and $Q\,$ of $0.5\,$ is shown below:

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

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

where cutoff frequency $F_c\,$ and $Q\,$ factor are discussed above in the low-pass filter discussion. The circuit above implements this transfer function by the equations

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

(as before), and

$Q = \frac{\sqrt{R_1R_2C_1C_2}}{R_2(C_1+C_2)}.$

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

[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 negative 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 $C 1 = C 2$ .

[edit] Implementation

The calculations above assume that all components used are ideal. This means, for example, that any kind of amplifier used in the implementation of the filter has infinite input impedance and no output impedance at any frequency of interest. It also means that all resistors and capacitors are exactly the values stated, with no tolerance for error. And it also means that the filter is driven by a signal source that has no impedance at any frequency of interest. Most of these assumptions are invalid in any actual implementation of these circuits because these ideal components do not exist. Therefore, the response of an actual filter will only approximate the theoretical response indicated by these calculations. How close this approximation is depends on how close the components utilized approximate the ideal.