Editing Sallen–Key topology (section)

==Generic Sallen–Key topology{{anchor|Generic Sallen–Key topology|Generic Sallen-Key topology}}==
[[File:Sallen-Key Generic Circuit.svg|thumb|upright=1.4|Figure&nbsp;1: The generic Sallen–Key filter topology]]
The generic unity-gain Sallen–Key filter topology implemented with a unity-gain operational amplifier is shown in Figure&nbsp;1. The following analysis is based on the assumption that the operational amplifier is ideal.

Because the op amp is in a [[negative feedback|negative-feedback]] configuration, its <math>v_+</math> and <math>v_-</math> inputs must match (i.e., <math>v_+ = v_-</math>). However, the inverting input <math>v_-</math> is connected directly to the output <math>v_\text{out}</math>, and so

{{NumBlk|:|<math>v_+ = v_- = v_\text{out}.</math>|1}}

By [[Kirchhoff's circuit laws|Kirchhoff's current law]] (KCL) applied at the <math>v_x</math> node,

{{NumBlk|:|<math>\frac{v_\text{in} - v_x}{Z_1} = \frac{v_x-v_\text{out}}{Z_3} + \frac{v_x - v_-}{Z_2}.</math>|2}}

By combining equations&nbsp;(1) and&nbsp;(2),

:<math>\frac{v_\text{in} - v_x}{Z_1} = \frac{v_x - v_\text{out}}{Z_3} + \frac{v_x - v_\text{out}}{Z_2}.</math>

Applying equation&nbsp;(1) and KCL at the op amp's non-inverting input <math>v_+</math> gives

:<math>\frac{v_x - v_\text{out}}{Z_2} = \frac{v_\text{out}}{Z_4},</math>

which means that

{{NumBlk|:|<math>v_x = v_\text{out} \left( \frac{Z_2}{Z_4} + 1 \right).</math>|3}}

Combining equations&nbsp;(2) and&nbsp;(3) gives

{{NumBlk|:|<math>\frac{v_\text{in} - v_\text{out} \left( \frac{Z_2}{Z_4} + 1 \right)}{Z_1} = \frac{v_\text{out} \left( \frac{Z_2}{Z_4} + 1 \right) - v_\text{out}}{Z_3} + \frac{v_\text{out} \left( \frac{Z_2}{Z_4} + 1 \right) - v_\text{out}}{Z_2}.</math>|4}}

Rearranging equation&nbsp;(4) gives the [[transfer function]]

{{NumBlk|:|<math>\frac{v_\text{out}}{v_\text{in}} = \frac{Z_3 Z_4}{Z_1 Z_2 + Z_3(Z_1 + Z_2) + Z_3 Z_4},</math>|5}}

which typically describes a second-order [[LTI system theory|linear time-invariant (LTI) system]].

If the <math>Z_3</math> component were connected to ground instead of to <math>v_\text{out}</math>, the filter would be a [[voltage divider]] composed of the <math>Z_1</math> and <math>Z_3</math> components cascaded with another voltage divider composed of the <math>Z_2</math> and <math>Z_4</math> components. The buffer amplifier [[Bootstrapping (electronics)|bootstraps]] the "bottom" of the <math>Z_3</math> 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 [[Passivity (engineering)|passive components]] (e.g., [[resistor]]s and [[capacitor]]s) for <math>Z_1</math>, <math>Z_2</math>, <math>Z_4</math>, and <math>Z_3</math>, the filter can be made with [[#Application:_low-pass_filter|low-pass]], [[#Application:_bandpass_filter|bandpass]], and [[#Application:_high-pass_filter|high-pass]] characteristics. In the examples below, recall that a resistor with [[electrical resistance|resistance]] <math>R</math> has [[electrical impedance|impedance]] <math>Z_R</math> of
:<math>Z_R = R,</math>
and a capacitor with [[capacitance]] <math>C</math> has impedance <math>Z_C</math> of
:<math>Z_C = \frac{1}{s C},</math>
where <math>s = j \omega = 2 \pi j f</math> (here <math>j</math> denotes the [[imaginary unit]]) is the [[complex number|complex]] [[angular frequency]], and <math>f</math> 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.