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| The '''Sallen–Key topology''' is an [[electronic filter topology]] used to implement [[Low-pass filter#Continuous-time low-pass filters|second-order]] [[active filter]]s that is particularly valued for its simplicity.<ref name="EE315A_Notes">[https://ccnet.stanford.edu/cgi-bin/course.cgi?cc=ee315a&action=handout_download&handout_id=ID126954294624704 "EE315A Course Notes - Chapter 2"-B. Murmann]</ref> It is a [[degeneracy (mathematics)|degenerate]] form of a '''voltage-controlled voltage-source (VCVS) filter topology'''. A VCVS filter uses a [[wiktionary:super-|super]]-[[1 (number)|unity]]-[[gain]] voltage amplifier with practically infinite [[input impedance]] and zero [[output impedance]] to implement a [[Pole (complex analysis)|2-pole]] (12 dB/octave) [[low-pass]], [[high-pass]], or [[bandpass]] [[frequency response|response]]. The super-unity-gain amplifier allows for very high [[Q factor]] and [[passband]] gain without the use of [[inductor]]s. A Sallen–Key filter is a variation on a VCVS filter that uses a unity-gain amplifier (i.e., a pure [[buffer amplifier]] with 0 [[Decibel|dB]] gain). It was introduced by [[R. P. Sallen]] and [[E. L. Key]] of [[MIT]] [[Lincoln Laboratory]] in 1955.<ref name="SallenKey">{{cite journal|title=A Practical Method of Designing RC Active Filters|journal=IRE Transactions on Circuit Theory|date=March 1955|first=R. P.|last=Sallen|coauthors=E. L. Key|volume=2|issue=1|pages=74–85|id= |url=|format=|accessdate=2007-08-14}}</ref>
| | They also have trained professional workers to deal with the problem more accurately. Water damage cleanup and mold remediation right after a deluge should be carried out by certified installers. Water damage is slow but it always has effects which last for a long time. replacing them with impact-resistant glass can not only offer you the. Failure to professionally clean, treat and mitigate water related damages may result in a reduction of selling price and increased insurance premiums for years to come. <br><br>One important thing in this regard is the removal of excess water from the floor, carpets and articles of furniture. Not only that but you will need to dry out the floor underneath of the carpet to keep the mold from forming. Step 6- Put It Back Together And Try To Turn It On - This is the final test to see if you have handled the Blackberry Curve water damage successfully. Insurance rates are based in part on how much has been paid out on claims, so when companies like that take more than they should be, they are contributing to higher insurance rates for everyone else. Replacement of pavers is reasonably easy to complete with attractive results since the drive is composed of individual stones. <br><br>You should always keep the local plumbing or water damage Restoration Company. The Dallas Water Damage Restoration firms normally have an emergency system, to support the public in this dire and hostile situation. This article outlines weather and how to prepare for the various conditions you may (or may not) be expecting on your holidays, rain, shine, and everything in between. You can also install insulating sleeves on the pipes themselves, including some that use electricity to keep the pipes warm on particularly cold nights. The process of drying, decreasing the humidity from the wet area, cleaning your home and disinfecting is already daunting just by imagining it. <br><br>Nobody intends to harm their Black - Berry, and the thought of damaging it in any way would bring most people out in a cold sweat. For example, imagine you are selling the finest Mercedes-Benz in the world. This type of water comes from dishwashers and clothes washers. Well that really depends on why your i - Pod went white. " Lay out valuable wet record on top of clean paper towels in a well-ventilated and dry area. <br><br>Industrial fans are the quickest method, but if those aren't available any fan or dryer can be used. Such mold spores normally don’t present a problem until they land on a damp surface where they feed off the material and reproduce, causing water damage mold to develop. Provide the staff a map of where water shut-off valves are located. When choosing a contractor to complete [http://michaldymi.livejournal.com water damage restoration] services, be sure the technicians are IICRC certified. You'll also want to examine the building materials that cover the floor and ceiling of your RV, such as the carpeting and wood panels, because water damage can cause the carpet to get moldy, and ruin the ceiling panels. |
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| Because of its high input impedance and easily selectable gain, an [[operational amplifier]] in a conventional [[Operational amplifier applications#Non-inverting amplifier|non-inverting configuration]] is often used in VCVS implementations.{{Citation needed|date=January 2009}} Implementations of Sallen–Key filters often use an operational amplifier configured as a [[Operational amplifier applications#Voltage follower|voltage follower]]; however, [[common collector|emitter]] or [[common drain|source]] followers are other common choices for the buffer amplifier.
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| VCVS filters are relatively resilient to component [[tolerance (engineering)|tolerance]], but obtaining high Q factor may require extreme component value spread or high amplifier gain.<ref name="EE315A_Notes" /> Higher-order filters can be obtained by cascading two or more stages.
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| ==Generic Sallen–Key topology{{anchor|Generic Sallen–Key topology|Generic Sallen-Key topology}}==
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| 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.
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| [[File:Sallen-Key Generic Circuit.svg|frame|right|Figure 1: The generic Sallen–Key filter topology.]]
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| Because the operational amplifier (OA) is in a [[negative feedback|negative-feedback]] configuration, its ''v''<sub>+</sub> and ''v''<sub>-</sub> inputs must match (i.e., ''v''<sub>+</sub> = ''v''<sub>-</sub>). However, the inverting input ''v''<sub>-</sub> is connected directly to
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| the output ''v''<sub>out</sub>, and so
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| :<div style="float:left;"><math>v_+ = v_- = v_{\text{out}}.\,</math></div><div style="text-align: right;"><math>(1)\,</math></div><br clear="left">
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| By [[Kirchhoff's circuit laws|Kirchhoff's current law]] (KCL) applied at the ''v''<sub>x</sub> node,
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| :<div style="float:left;"><math>\frac{v_{\text{in}}-v_x}{Z_1}=\frac{v_x-v_{\text{out}}}{Z_3}+\frac{v_x-v_-}{Z_2}.</math></div><div style="text-align:right;"><math>(2)\,</math></div><br clear="left">
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| By combining Equations (1) and (2),
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| :<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>
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| Applying Equation (1) and KCL at the OA's non-inverting input ''v''<sub>+</sub> gives
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| :<math>\frac{v_x-v_{\text{out}}}{Z_2}=\frac{v_{\text{out}}}{Z_4},</math>
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| which means that
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| :<div style="float:left;"><math>v_x=v_{\text{out}} \left( \frac{Z_2}{Z_4}+1 \right).</math></div><div style="text-align:right;"><math>(3)\,</math></div><br clear="left">
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| Combining Equations (2) and (3) gives
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| :<div style="float:left;"><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></div><div style="text-align:right;"><math>(4)\,</math></div><br clear="left">
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| Rearranging Equation (4) gives the [[transfer function]]
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| :<div style="float:left;"><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></div><div style="text-align:right;"><math>(5)\,</math></div><br clear="left">
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| which typically describes a second-order [[LTI system theory|LTI system]].
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| ===Interpretation===
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| If the <math>Z_3\,</math> component were connected to ground, 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 [[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 operational amplifier's non-inverting input below the inverting input, thus emphasizing the similarity between the output and ground.
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| ===Branch impedances===
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| 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_\mathrm{R}\,</math> of
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| :<math>Z_\mathrm{R} = R\,,</math>
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| and a capacitor with [[capacitance]] <math>C\,</math> has impedance <math>Z_\mathrm{C}\,</math> of
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| :<math>Z_\mathrm{C} = \frac{1}{s C}\,,</math>
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| where <math>s = j \omega = \left(\sqrt{-1}\right) 2 \pi f\,</math> 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.
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| ==Application: Low-pass filter{{anchor|Low-pass configuration}}==
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| An example of a unity-gain low-pass configuration is shown in Figure 2.
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| [[File:Sallen-Key Lowpass General.svg|frame|right|Figure 2: A unity-gain low-pass filter implemented with a Sallen–Key topology.]]
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| 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
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| :<math>
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| Z_1 = R_1, \quad Z_2 = R_2, \quad Z_3 = \frac{1}{s C_1}, \quad \text{and} \quad Z_4 = \frac{1}{s C_2}.\,
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| </math>
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| The [[transfer function]] for this second-order unity-gain low-pass filter is
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| :<math>H(s) = { {\omega_0}^2 \over s^2 + 2 \alpha s + {\omega_0}^2 } </math>
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| where the [[Damping|undamped natural frequency]] <math>f_0\,</math>, [[attenuation coefficient|attenuation]] <math>\alpha</math>, and [[Q factor]] <math>Q\,</math> (i.e., [[damping ratio]] <math>\zeta</math>) are given by
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| :<math> \omega_0 = 2 \pi f_0 = \frac{1}{ \sqrt{R_1R_2C_1C_2} } </math>
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| and
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| :<math> 2 \alpha = 2 \zeta \omega_0 = \frac{\omega_0}{Q} = \frac{1}{C_1} \left( \frac{1}{R_1} + \frac{1}{R_2} \right)
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| = { 1 \over C_1 } \left( { R_1 + R_2 \over R_1 R_2 } \right) .\,</math>
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| So,
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| :<math> Q = { \omega_0 \over 2 \alpha } = \frac{ \sqrt{ R_1 R_2 C_1 C_2 } }{ C_2 \left( R_1 + R_2 \right) }
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| \qquad</math>
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| The <math>Q\,</math> 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 [[resonance|resonant]] [[frequency]] near <math>f_0\,</math> (see "[[LC filter]]" for a related discussion).
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| ===Poles and zeros===
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| This transfer function has no zeros and two [[pole-zero plot|poles]] located in the complex [[s plane|''s''-plane]]:
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| :<math> s = -\alpha \pm \sqrt{ \alpha^2 - {\omega_0}^2 } </math>
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| === Design choices===
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| A [[filter design|designer]] must choose the <math>Q\,</math> and <math>f_0\,</math> appropriate for their application.
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| The <math>Q</math> value is critical in determining the eventual shape. | |
| For example, a second-order [[Butterworth filter]], which has maximally flat passband frequency response, has a <math>Q\,</math> of <math>1/\sqrt{2}\,</math>.
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| By comparison, a value of <math>Q=1/2</math> corresponds to the series of two identical simple low-pass filters.
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| 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 <math>C_1\,</math> and <math>C_2\,</math> as <math>n\,</math> and the ratio between <math>R_1\,</math> and <math>R_2\,</math> as <math>m\,</math>. So,
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| :<math>R_1=mR,\,</math>
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| :<math>R_2=R,\,</math>
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| :<math>C_1=nC,\,</math>
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| :<math>C_2=C.\,</math>
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| Therefore, the <math>f_0\,</math> and <math>Q\,</math> expressions are
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| :<math> \omega_0 = 2 \pi f_0 = \frac{1}{RC\sqrt{mn}},\, </math>
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| and
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| :<math> Q = \frac{\sqrt{mn}}{m+1}. </math>
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| [[File:Sallen-Key Lowpass Example.svg|frame|Figure 3: A low-pass filter, which is implemented with a Sallen–Key topology, with ''f''<sub>c</sub>=15.9 kHz and ''Q'' = 0.5.]]
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| In practice, certain choices of component values will perform better than others due to the non-idealities of real operational amplifiers.<ref>[http://www.ti.com/lit/an/slyt306/slyt306.pdf Stop-band limitations of the Sallen-Key low-pass filter]</ref>
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| ===Example===
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| For example, the circuit in Figure 3 has an <math>f_0\,</math> of <math>15.9\,\text{kHz}\,</math> and a <math>Q\,</math> of <math>0.5\,</math>. The [[transfer function]] is given by | |
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| :<math>H(s)=\frac{1}{1+\underbrace{C_2(R_1+R_2)}_{\frac{2 \zeta}{\omega_0} = \frac{1}{\omega_0 Q} }s+\underbrace{C_1C_2R_1R_2}_{\frac{1}{{\omega_0}^2}}s^2},</math>
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| and, after substitution, this expression is equal to
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| :<math>H(s)=\frac{1}{1+\underbrace{RC(m+1)}_{\frac{2 \zeta}{\omega_0} = \frac{1}{\omega_0 Q} }s+\underbrace{mnR^2C^2}_{\frac{1}{{\omega_0}^2}}s^2}</math>
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| which shows how every <math>(R,C)\,</math> combination comes with some <math>(m,n)\,</math> combination to provide the same <math>f_0</math> and <math>Q</math> for the low-pass filter. A similar design approach is used for the other filters below.
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| ==Application: High-pass filter==
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| [[File:Sallen-Key Highpass Example.svg|frame|right|Figure 4: A specific Sallen–Key high-pass filter with ''f''<sub>c</sub>=72 Hz and ''Q'' = 0.5.]]
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| A second-order unity-gain high-pass filter with <math>f_0\,</math> of <math>72\,\text{Hz}\,</math> and <math>Q\,</math> of <math>0.5\,</math> is shown in Figure 4.
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| A second-order unity-gain high-pass filter has the transfer function
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| :<math> H(s) = \frac{s^2}{s^2+\underbrace{2\pi\left(\frac{f_0}{Q}\right)}_{2 \zeta \omega_0 = \frac{\omega_0}{Q}}s+\underbrace{(2\pi f_0)^2}_{{\omega_0}^2}}, </math>
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| where undamped natural frequency <math>f_0\,</math> and <math>Q\,</math> factor are discussed above in the [[#Low-pass configuration|low-pass filter]] discussion. The circuit above implements this transfer function by the equations
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| :<math> \omega_0 = 2 \pi f_0 = \frac{1}{\sqrt{R_1R_2C_1C_2}}\,</math>
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| (as before), and
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| :<math> \frac{1}{2\zeta} = Q = { \omega_0 \over 2 \alpha } = \frac{\sqrt{R_1R_2C_1C_2}}{R_1(C_1+C_2)}.\,</math>
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| So
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| :<math> 2 \alpha = 2 \zeta \omega_0 = \frac{ \omega_0 }{ Q } = \frac{ C_1 + C_2 }{ R_2 C_1 C_2 }. </math>
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| Follow an approach similar to the one used to design the low-pass filter above.
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| == Application: Bandpass filter==
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| [[File:VCVS Filter Bandpass General.svg|frame|right|Figure 5: A bandpass filter realized with a VCVS topology.]]
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| 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|generic Sallen–Key topology]]. Its transfer function is given by:
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| :<math>H(s) = \frac{\overbrace{\left(1+\frac{R_\mathrm{b}}{R_\mathrm{a}}\right)}^{G} \frac{s}{R_1 C_1}}{s^2 + | |
| \underbrace{\left( \frac{1}{R_1 C_1} + \frac{1}{R_2 C_1} + \frac{1}{R_2 C_2} - \frac{R_\mathrm{b}}{R_\mathrm{a} R_\mathrm{f} C_1} \right)}_{2 \zeta \omega_0 = \frac{\omega_0}{Q}} s +
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| \underbrace{\frac{R_1 + R_\mathrm{f}}{R_1 R_\mathrm{f} R_2 C_1 C_2}}_{{\omega_0}^2 = (2\pi f_0)^2}}</math>
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| The [[center frequency]] <math>f_0</math> (i.e., the frequency where the magnitude response has its ''peak'') is given by:
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| :<math> f_0=\frac{1}{2\pi}\sqrt{\frac{R_\mathrm{f}+R_1}{C_1C_2R_1R_2R_\mathrm{f}}} </math>
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| The [[Q factor]] <math>Q</math> is given by
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| :<math> \begin{align} Q
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| &= \frac{\omega_0}{2 \zeta \omega_0}
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| = \frac{\omega_0}{\frac{\omega_0}{Q}}\\
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| &= \frac{\sqrt{\frac{R_1 + R_\mathrm{f}}{R_1 R_\mathrm{f} R_2 C_1 C_2}}}{ \frac{1}{R_1 C_1} + \frac{1}{R_2 C_1} + \frac{1}{R_2 C_2} - \frac{R_\mathrm{b}}{R_\mathrm{a} R_\mathrm{f} C_1} }\\
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| &= \frac{\sqrt{ (R_1 + R_\mathrm{f}) R_1 R_\mathrm{f} R_2 C_1 C_2}}{ R_1 R_\mathrm{f} (C_1 + C_2) + R_2 C_2 ( R_\mathrm{f} - \frac{R_\mathrm{b}}{R_\mathrm{a}} R_1 ) }
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| \end{align}</math>
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| The voltage divider in the negative feedback loop controls the gain. The "inner gain" <math>G</math> provided by the operational amplifier is given by
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| :<math> G=1+\frac{R_\mathrm{b}}{R_\mathrm{a}}</math>
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| while the amplifier gain at the peak frequency is given by:
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| :<math> A=\frac{G}{3-G}</math>
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| It can be seen that <math>G</math> must be kept below 3 or else the filter will oscillate.
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| ==See also==
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| * [[Filter design]]
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| * [[Electronic filter topology]]
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| * [[Damping]]
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| * [[Harmonic oscillator]]
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| * [[Resonance]]
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| ==References==
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| {{reflist}}
| |
| | |
| ==External links==
| |
| * [http://focus.ti.com/lit/an/sloa024b/sloa024b.pdf Texas Instruments Application Report: Analysis of the Sallen–Key Architecture]
| |
| * [http://www.analog.com/en/amplifiers-and-comparators/products/dt-adisim-design-sim-tool/Filter_Wizard/resources/fca.html Analog Devices filter design applet] – A simple online tool for designing active filters using voltage-feedback op-amps.
| |
| * [http://www-k.ext.ti.com/SRVS/CGI-BIN/WEBCGI.EXE/,/?St=147,E=0000000000002472277,K=2597,Sxi=1,Case=obj(26717) TI active filter design source FAQ]
| |
| * [http://focus.ti.com/lit/ml/sloa088/sloa088.pdf Op Amps for Everyone – Chapter 16]
| |
| * [http://postreh.com/vmichal/papers/frequency_filters_with_high_attenuation_Radio2009VMJS.pdf High frequency modification of Sallen-Key filter - improving the stopband attenuation floor]
| |
| * [http://www.changpuak.ch/electronics/calc_08.php Online Calculation Tool for Sallen–Key Low-pass/High-pass Filters]
| |
| * [http://sim.okawa-denshi.jp/en/Fkeisan.htm Online Calculation Tool for Filter Design and Analysis]
| |
| * [http://www.tedpavlic.com/teaching/osu/ece327/lab7_proj/lab7_proj_procedure.pdf ECE 327: Procedures for Output Filtering Lab] – Section 3 ("Smoothing Low-Pass Filter") discusses active filtering with Sallen–Key Butterworth low-pass filter.
| |
| * [http://www.youtube.com/watch?v=DEV7l66D6Ys Filtering 101: Multi Pole Filters with Sallen-Key], Matt Duff of Analog Devices explains how Sallen Key circuit works
| |
| | |
| {{DEFAULTSORT:Sallen-Key Topology}}
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| [[Category:Linear filters]]
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| [[Category:Electronic filter topology]]
| |
They also have trained professional workers to deal with the problem more accurately. Water damage cleanup and mold remediation right after a deluge should be carried out by certified installers. Water damage is slow but it always has effects which last for a long time. replacing them with impact-resistant glass can not only offer you the. Failure to professionally clean, treat and mitigate water related damages may result in a reduction of selling price and increased insurance premiums for years to come.
One important thing in this regard is the removal of excess water from the floor, carpets and articles of furniture. Not only that but you will need to dry out the floor underneath of the carpet to keep the mold from forming. Step 6- Put It Back Together And Try To Turn It On - This is the final test to see if you have handled the Blackberry Curve water damage successfully. Insurance rates are based in part on how much has been paid out on claims, so when companies like that take more than they should be, they are contributing to higher insurance rates for everyone else. Replacement of pavers is reasonably easy to complete with attractive results since the drive is composed of individual stones.
You should always keep the local plumbing or water damage Restoration Company. The Dallas Water Damage Restoration firms normally have an emergency system, to support the public in this dire and hostile situation. This article outlines weather and how to prepare for the various conditions you may (or may not) be expecting on your holidays, rain, shine, and everything in between. You can also install insulating sleeves on the pipes themselves, including some that use electricity to keep the pipes warm on particularly cold nights. The process of drying, decreasing the humidity from the wet area, cleaning your home and disinfecting is already daunting just by imagining it.
Nobody intends to harm their Black - Berry, and the thought of damaging it in any way would bring most people out in a cold sweat. For example, imagine you are selling the finest Mercedes-Benz in the world. This type of water comes from dishwashers and clothes washers. Well that really depends on why your i - Pod went white. " Lay out valuable wet record on top of clean paper towels in a well-ventilated and dry area.
Industrial fans are the quickest method, but if those aren't available any fan or dryer can be used. Such mold spores normally don’t present a problem until they land on a damp surface where they feed off the material and reproduce, causing water damage mold to develop. Provide the staff a map of where water shut-off valves are located. When choosing a contractor to complete water damage restoration services, be sure the technicians are IICRC certified. You'll also want to examine the building materials that cover the floor and ceiling of your RV, such as the carpeting and wood panels, because water damage can cause the carpet to get moldy, and ruin the ceiling panels.