Sallen–Key topology: Difference between revisions

The Sallen–Key topology is an electronic filter topology used to implement second-order active filters that is particularly valued for its simplicity . It is a degenerate form of a voltage-controlled voltage-source (VCVS) filter topology. A VCVS filter uses a super-unity-gain voltage amplifier with practically infinite input impedance and zero output impedance to implement a 2-pole (12 dB/octave) low-pass, high-pass, or bandpass response. The super-unity-gain amplifier allows for very high Q factor and passband gain without the use of inductors. 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 dB gain). It was introduced by R.P. Sallen and E. L. Key of MIT Lincoln Laboratory in 1955.

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 operational amplifier configured as a voltage follower; however, emitter or source followers are other common choices for the buffer amplifier.

VCVS filters are relatively resilient to component tolerance, 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.

Figure 1: The generic Sallen–Key filter topology.

Because the operational amplifier (OA) is in a negative-feedback configuration, its v₊ and v_- inputs must match (i.e., v₊ = v_-). However, the inverting input v_- is connected directly to the output v_out, and so

$v_{+}=v_{-}=v_{\text{out}}.\,$

$(1)\,$

By Kirchhoff's current law (KCL) applied at the v_x node,

${\frac {v_{\text{in}}-v_{x}}{Z_{1}}}={\frac {v_{x}-v_{\text{out}}}{Z_{4}}}+{\frac {v_{x}-v_{-}}{Z_{2}}}.$

$(2)\,$

By combining Equations (1) and (2),

${\frac {v_{\text{in}}-v_{x}}{Z_{1}}}={\frac {v_{x}-v_{\text{out}}}{Z_{4}}}+{\frac {v_{x}-v_{\text{out}}}{Z_{2}}}$

Applying KCL again at the OA's non-inverting input v₊ (= v_- = v_out) gives

${\frac {v_{x}-v_{\text{out}}}{Z_{2}}}={\frac {v_{\text{out}}}{Z_{3}}},$

which means that

$v_{x}=v_{\text{out}}\left({\frac {Z_{2}}{Z_{3}}}+1\right).$

$(3)\,$

Combining Equations (2) and (3) gives

${\frac {v_{\text{in}}-v_{\text{out}}\left({\frac {Z_{2}}{Z_{3}}}+1\right)}{Z_{1}}}={\frac {v_{\text{out}}\left({\frac {Z_{2}}{Z_{3}}}+1\right)-v_{\text{out}}}{Z_{4}}}+{\frac {v_{\text{out}}\left({\frac {Z_{2}}{Z_{3}}}+1\right)-v_{\text{out}}}{Z_{2}}}.$

$(4)\,$

Rearranging Equation (4) gives the transfer function

${\frac {v_{\text{out}}}{v_{\text{in}}}}={\frac {Z_{3}Z_{4}}{Z_{1}Z_{2}+Z_{4}(Z_{1}+Z_{2})+Z_{3}Z_{4}}},$

$(5)\,$

which typically describes a second-order LTI system.

Interpretation

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.

Example applications

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. In the 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

$Z_{C}={\frac {1}{sC}}\,,$

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.

Example: Low-pass filter

An example of a unity-gain low-pass configuration is shown in Figure 2.

Figure 2: A unity-gain low-pass filter implemented with a Sallen–Key topology.

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

$Z_{1}=R_{1},\quad Z_{2}=R_{2},\quad Z_{3}={\frac {1}{sC_{2}}},\quad {\text{and}}\quad Z_{4}={\frac {1}{sC_{1}}}.\,$

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

$H(s)={\frac {\overbrace {(2\pi f_{c})^{2}} ^{\omega _{c}^{2}}}{s^{2}+\underbrace {2\pi {\frac {f_{c}}{Q}}} _{{\frac {\omega _{c}}{Q}}=2\zeta \omega _{c}}s+\underbrace {(2\pi f_{c})^{2}} _{\omega _{c}^{2}}}}$

where the cutoff frequency $f_{c}\,$ and Q factor $Q\,$ (i.e., damping ratio $\zeta$) are given by

$f_{c}={\frac {1}{2\pi {\sqrt {R_{1}R_{2}C_{1}C_{2}}}}}$

and

$2\zeta ={\frac {1}{Q}}={\frac {\sqrt {R_{1}R_{2}C_{1}C_{2}}}{C_{1}}}\left({\frac {1}{R_{1}}}+{\frac {1}{R_{2}}}\right).\,$

So,

$Q={\frac {\sqrt {R_{1}R_{2}C_{1}C_{2}}}{C_{2}\left(R_{1}+R_{2}\right)}}\qquad$

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

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

Figure 3: A low-pass filter, which is implemented with a Sallen–Key topology, with f_c=15.9 kHz and Q = 0.5.

For example, the circuit in Figure 3 has an $f_{c}\,$ of $15.9\,{\text{kHz}}\,$ and a $Q\,$ of $0.5\,$. The transfer function is given by

$H(s)={\frac {1}{1+\underbrace {C_{2}(R_{1}+R_{2})} _{{\frac {2\zeta }{\omega _{c}}}={\frac {1}{\omega _{c}Q}}}s+\underbrace {C_{1}C_{2}R_{1}R_{2}} _{\frac {1}{\omega _{c}^{2}}}s^{2}}},$

and, after substitution, this expression is equal to

$H(s)={\frac {1}{1+\underbrace {RC(m+1)} _{{\frac {2\zeta }{\omega _{c}}}={\frac {1}{\omega _{c}Q}}}s+\underbrace {mnR^{2}C^{2}} _{\frac {1}{\omega _{c}^{2}}}s^{2}}}$

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.

Example: High-pass filter

Figure 4: A specific Sallen–Key high-pass filter with f_c=72 Hz and Q = 0.5.

A second-order unity-gain high-pass filter with $f_{c}\,$ of $72\,{\text{Hz}}\,$ and $Q\,$ of $0.5\,$ is shown in Figure 4.

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

$H(s)={\frac {s^{2}}{s^{2}+\underbrace {2\pi ({\frac {f_{c}}{Q}})} _{2\zeta \omega _{c}={\frac {\omega _{c}}{Q}}}s+\underbrace {(2\pi f_{c})^{2}} _{\omega _{c}^{2}}}},$

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

$f_{c}={\frac {1}{2\pi {\sqrt {R_{1}R_{2}C_{1}C_{2}}}}}\,$

(as before), and

${\frac {1}{2\zeta }}=Q={\frac {\sqrt {R_{1}R_{2}C_{1}C_{2}}}{R_{1}(C_{1}+C_{2})}}.\,$

So

$2\zeta f_{c}={\frac {f_{c}}{Q}}={\frac {C_{1}+C_{2}}{2\pi R_{2}C_{1}C_{2}}}.$

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

VCVS Example: Bandpass configuration

Figure 5: A bandpass filter realized with a VCVS topology.

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:

$H(s)={\frac {\overbrace {\left(1+{\frac {R_{b}}{R_{a}}}\right)} ^{G}{\frac {s}{R_{1}C_{1}}}}{s^{2}+\underbrace {\left({\frac {1}{R_{1}C_{1}}}+{\frac {1}{R_{2}C_{1}}}+{\frac {1}{R_{2}C_{2}}}-{\frac {R_{b}}{R_{a}R_{f}C_{1}}}\right)} _{2\zeta \omega _{0}={\frac {\omega _{0}}{Q}}}s+\underbrace {\frac {R_{1}+R_{f}}{R_{1}R_{f}R_{2}C_{1}C_{2}}} _{\omega _{0}^{2}=(2\pi f_{0})^{2}}}}$

The center frequency $f_{0}$ (i.e., the frequency where the magnitude response has its peak) is given by:

$f_{0}={\frac {1}{2\pi }}{\sqrt {\frac {R_{f}+R_{1}}{C_{1}C_{2}R_{1}R_{2}R_{f}}}}$

The voltage divider in the negative feedback loop controls the gain. The "inner gain" $G$ provided by the operational amplifier 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 $R_{2}=2R_{1}$ and $C_{1}=C_{2}$.

See also

• Filter design
• Electronic filter topology

External links

• Texas Instruments Application Report: Analysis of the Sallen–Key Architecture
• Analog Devices filter design applet – 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
• Real properties of Sallen–Key basic block
• Online Calculation Tool for Sallen–Key Low-pass/High-pass Filters
• ECE 327: Procedures for Output Filtering Lab – Section 3 ("Smoothing Low-Pass Filter") discusses active filtering with Sallen–Key Butterworth low-pass filter.