VCVS Filter
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The Sallen–Key topology is an electronic filter topology used to implement
second-order Second-order may refer to: Mathematics * Second order approximation, an approximation that includes quadratic terms * Second-order arithmetic, an axiomatization allowing quantification of sets of numbers * Second-order differential equation, a di ...
active filters that is particularly valued for its simplicity."EE315A Course Notes - Chapter 2"-B. Murmann
It is a
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form of a voltage-controlled voltage-source (VCVS) filter topology.


Explanation of operation

A VCVS filter uses a voltage amplifier with practically infinite input impedance and zero
output impedance The output impedance of an electrical network is the measure of the opposition to current flow (impedance), both static ( resistance) and dynamic ( reactance), into the load network being connected that is ''internal'' to the electrical source. The ...
to implement a 2-pole low-pass,
high-pass A high-pass filter (HPF) is an electronic filter that passes signals with a frequency higher than a certain cutoff frequency and attenuates signals with frequencies lower than the cutoff frequency. The amount of attenuation for each frequency de ...
,
bandpass A band-pass filter or bandpass filter (BPF) is a device that passes frequencies within a certain range and rejects (attenuates) frequencies outside that range. Description In electronics and signal processing, a filter is usually a two-port ...
,
bandstop In signal processing, a band-stop filter or band-rejection filter is a filter that passes most frequencies unaltered, but attenuates those in a specific range to very low levels. It is the opposite of a band-pass filter. A notch filter is a b ...
, or
allpass An all-pass filter is a signal processing filter that passes all frequencies equally in gain, but changes the phase relationship among various frequencies. Most types of filter reduce the amplitude (i.e. the magnitude) of the signal applied to it ...
response Response may refer to: *Call and response (music), musical structure *Reaction (disambiguation) *Request–response **Output (computing), Output or response, the result of telecommunications input *Response (liturgy), a line answering a versicle ...
. The VCVS filter allows high
Q factor In physics and engineering, the quality factor or ''Q'' factor is a dimensionless parameter that describes how underdamped an oscillator or resonator is. It is defined as the ratio of the initial energy stored in the resonator to the energy los ...
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-voltage-gain amplifier (i.e., a pure buffer amplifier). It was introduced by
R. P. Sallen R. or r. may refer to: * '' Reign'', the period of time during which an Emperor, king, queen, etc., is ruler. * '' Rex'', abbreviated as R., the Latin word meaning King * ''Regina'', abbreviated as R., the Latin word meaning Queen * or , abbrevi ...
and
E. L. Key E is the fifth letter of the Latin alphabet. E or e may also refer to: Commerce and transportation * €, the symbol for the euro, the European Union's standard currency unit * ℮, the estimated sign, an EU symbol indicating that the wei ...
of MIT
Lincoln Laboratory The MIT Lincoln Laboratory, located in Lexington, Massachusetts, is a United States Department of Defense federally funded research and development center chartered to apply advanced technology to problems of national security. Research and dev ...
in 1955.


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 Tolerance or toleration is the state of tolerating, or putting up with, conditionally. Economics, business, and politics * Toleration Party, a historic political party active in Connecticut * Tolerant Systems, the former name of Veritas Software ...
, but obtaining high Q factor may require extreme component value spread or high amplifier gain. Higher-order filters can be obtained by cascading two or more stages.


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 op amp is in a
negative-feedback Negative feedback (or balancing feedback) occurs when some function of the output of a system, process, or mechanism is fed back in a manner that tends to reduce the fluctuations in the output, whether caused by changes in the input or by other ...
configuration, its v_+ and v_- inputs must match (i.e., v_+ = v_-). However, the inverting input v_- is connected directly to the output v_\text, and so By Kirchhoff's current law (KCL) applied at the v_x node, By combining equations (1) and (2), :\frac = \frac + \frac. Applying equation (1) and KCL at the op amp's non-inverting input v_+ gives :\frac = \frac, which means that Combining equations (2) and (3) gives Rearranging equation (4) gives the transfer function which typically describes a second-order linear time-invariant (LTI) system. If the Z_3 component were connected to ground instead of to v_\text, 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 amplifier 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 op amp's non-inverting input below the inverting input, thus emphasizing the similarity between the output and ground.


Branch impedances

By choosing different
passive components Passivity is a property of engineering systems, most commonly encountered in analog electronics and control systems. Typically, analog designers use ''passivity'' to refer to incrementally passive components and systems, which are incapable of p ...
(e.g.,
resistor A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active el ...
s and capacitors) for Z_1, Z_2, Z_4, and Z_3, the filter can be made with low-pass,
bandpass A band-pass filter or bandpass filter (BPF) is a device that passes frequencies within a certain range and rejects (attenuates) frequencies outside that range. Description In electronics and signal processing, a filter is usually a two-port ...
, and
high-pass A high-pass filter (HPF) is an electronic filter that passes signals with a frequency higher than a certain cutoff frequency and attenuates signals with frequencies lower than the cutoff frequency. The amount of attenuation for each frequency de ...
characteristics. In the examples below, recall that a resistor with
resistance Resistance may refer to: Arts, entertainment, and media Comics * Either of two similarly named but otherwise unrelated comic book series, both published by Wildstorm: ** ''Resistance'' (comics), based on the video game of the same title ** ''T ...
R has impedance Z_R of :Z_R = R, and a capacitor with capacitance C has impedance Z_C of :Z_C = \frac, where s = j \omega = 2 \pi j f (here j denotes the imaginary unit) is the complex angular frequency, and f is the frequency of a pure
sine-wave A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ma ...
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 :Z_1 = R_1, \quad Z_2 = R_2, \quad Z_3 = \frac, \quad Z_4 = \frac. The transfer function for this second-order unity-gain low-pass filter is :H(s) = \frac, where the undamped natural frequency f_0, attenuation \alpha, Q factor Q, and
damping ratio Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples inc ...
\zeta, are given by : \omega_0 = 2 \pi f_0 = \frac and : 2 \alpha = 2 \zeta \omega_0 = \frac = \frac \left( \frac + \frac \right) = \frac \left( \frac \right). So, : Q = \frac = \frac. 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 Resonance describes the phenomenon of increased amplitude that occurs when the frequency of an applied periodic force (or a Fourier component of it) is equal or close to a natural frequency of the system on which it acts. When an oscillatin ...
near f_0 (see "
LC filter An LC circuit, also called a resonant circuit, tank circuit, or tuned circuit, is an electric circuit consisting of an inductor, represented by the letter L, and a capacitor, represented by the letter C, connected together. The circuit can ac ...
" for a related discussion).


Poles and zeros

This transfer function has no (finite) zeros and two
poles Poles,, ; singular masculine: ''Polak'', singular feminine: ''Polka'' or Polish people, are a West Slavic nation and ethnic group, who share a common history, culture, the Polish language and are identified with the country of Poland in Ce ...
located in the complex ''s''-plane: : s = -\alpha \pm \sqrt. 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 The Butterworth filter is a type of signal processing filter designed to have a frequency response that is as flat as possible in the passband. It is also referred to as a maximally flat magnitude filter. It was first described in 1930 by the Br ...
, which has maximally flat passband frequency response, has a Q of 1/\sqrt. 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 the ratio between both resistors as well as that between the capacitors. One possibility is to set the ratio between C_1 and C_2 as n versus 1/n and the ratio between R_1 and R_2 as m versus 1/m. So, : \begin R_1 &= mR, \\ R_2 &= R/m, \\ C_1 &= nC, \\ C_2 &= C/n. \end As a result, the f_0 and Q expressions are reduced to : \omega_0 = 2 \pi f_0 = \frac and : Q = \frac. Starting with a more or less arbitrary choice for e.g. C and n, the appropriate values for R and m can be calculated in favor of the desired f_0 and Q. In practice, certain choices of component values will perform better than others due to the non-idealities of real operational amplifiers.Stop-band limitations of the Sallen–Key low-pass filter
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 f_0 = 15.9~\text and Q = 0.5. The transfer function is given by :H(s) = \frac, and, after substitution, this expression is equal to :H(s) = \frac, 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 :Z(s) = R_1\frac, where s' = \frac and k = \frac = \frac. Furthermore, for Q>\sqrt, there is a minimal value of the magnitude of the impedance, given by Eq. (16) of Cartwright and Kaminsky, which states that :, Z(s), _\text = R_1\sqrt. Fortunately, this equation is well-approximated by :, Z(s), _\text \approx R_1\sqrt for 0.25\leq k \leq 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., :\omega_\text = \omega_0\sqrt. This equation can also be well approximated using Eq. (20) of Cartwright and Kaminsky, which states that :\omega_\text \approx \omega_0\sqrt.


Application: high-pass filter

A second-order unity-gain high-pass filter with f_0 = 72~\text and Q = 0.5 is shown in Figure 4. A second-order unity-gain high-pass filter has the transfer function : H(s) = \frac, 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 : \omega_0 = 2 \pi f_0 = \frac (as before) and : \frac = Q = \frac = \frac. So : 2\alpha = 2\zeta\omega_0 = \frac = \frac. 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 :H(s) = \frac. The center frequency f_0 (i.e., the frequency where the magnitude response has its ''peak'') is given by : f_0 = \frac\sqrt. The Q factor Q is given by : \begin Q &= \frac = \frac\\
0pt PT, Pt, or pt may refer to: Arts and entertainment * ''P.T.'' (video game), acronym for ''Playable Teaser'', a short video game released to promote the cancelled video game ''Silent Hills'' * Porcupine Tree, a British progressive rock group ...
&= \frac\\
0pt PT, Pt, or pt may refer to: Arts and entertainment * ''P.T.'' (video game), acronym for ''Playable Teaser'', a short video game released to promote the cancelled video game ''Silent Hills'' * Porcupine Tree, a British progressive rock group ...
&= \frac. \end The voltage divider in the negative feedback loop controls the "inner gain" G of the op amp: : G = 1 + \frac. If the inner gain G is too high, the filter will oscillate.


See also

* Filter design * Electronic filter topology *
Harmonic oscillator In classical mechanics, a harmonic oscillator is a system that, when displaced from its Mechanical equilibrium, equilibrium position, experiences a restoring force ''F'' Proportionality (mathematics), proportional to the displacement ''x'': \v ...
* Resonance


References


External links


Texas Instruments Application Report: Analysis of the Sallen–Key Architecture

Analog Devices filter design tool
nbsp;– A simple online tool for designing active filters using voltage-feedback op-amps.
TI active filter design source FAQ

Op Amps for Everyone – Chapter 16

High frequency modification of Sallen-Key filter - improving the stopband attenuation floor

Online Calculation Tool for Sallen–Key Low-pass/High-pass Filters



ECE 327: Procedures for Output Filtering Lab
nbsp;– Section 3 ("Smoothing Low-Pass Filter") discusses active filtering with Sallen–Key Butterworth low-pass filter.
Filtering 101: Multi Pole Filters with Sallen-Key
Matt Duff of Analog Devices explains how Sallen Key circuit works {{DEFAULTSORT:Sallen-Key Topology Linear filters Electronic filter topology