Mini Tutorial MT-222 One Technology Way • P.O. Box 9106 • Norwood, MA 02062-9106, U.S.A. • Tel: 781.329.4700 • Fax: 781.461.3113 • www.analog.com While the Sallen-Key filter is widely used, a serious drawback is that the filter is not easily tuned, due to interaction of the component values on F0 and Q. Another limitation is the relatively low maximum Q value obtainable. by Hank Zumbahlen, Analog Devices, Inc. IN THIS MINI TUTORIAL Three sample Sallen-Key filters are designed in this mini tutorial, one in a series of mini tutorials describing discrete circuits for precision op amps. The Sallen-Key configuration, also known as a voltage control voltage source (VCVS), was first introduced in 1955 by R. P. Sallen and E. L. Key of MIT’s Lincoln Labs (see the References section). One of the most widely used filter topologies, this configuration is shown in Figure 1. IN To transform the low pass into the highpass, simply exchange the capacitors and the resistors in the frequency determining network (that is, not the amp gain resistors). This is shown in Figure 2. The comments regarding sensitivity of the filter given above for the low-pass case apply to the high-pass case as well. The design equations for the Sallen-Key high-pass filter are shown in the Sallen-Key High-Pass Design Equations section. C1 R1 IN R2 R3 C1 R1 OUT C2 OUT 10427-002 Sallen-Key Filters R4 R2 Figure 2. Sallen-Key High-Pass Filter R4 10427-001 R3 Figure 1. Sallen-Key Low-Pass Filter One reason for this popularity is that this configuration shows the least dependence of filter performance on the performance of the op amp. This is because the op amp is configured as an amplifier, as opposed to an integrator, which minimizes the gain-bandwidth requirements of the op amp. This infers that for a given op amp, one can design a higher frequency filter than with other topologies since the op amp gain-bandwidth product does not limit the performance of the filter as it would if it were configured as an integrator. In addition, since the op amp is configured as an amplifier, current feedback amplifiers, which cannot be configured as conventional integrators, can be used. This allows slightly more bandwidth from the filter. The signal phase through the filter is maintained (noninverting configuration). The band-pass case of the Sallen-Key filter (see Figure 4) has a severe limitation. The value of Q determines the gain of the filter, that is, it cannot be set independently, as it can with the low-pass or high-pass cases. The design equations for the Sallen-Key band-pass filter are shown in the Sallen-Key BandPass Design Equations section. Although a Sallen-Key notch filter may also be constructed, notch filters have a large number of undesirable characteristics. The resonant frequency, or the notch frequency, cannot be adjusted easily due to component interaction. As in the band-pass case, the section gain is fixed by the other design parameters, and there is a wide spread in component values, especially capacitors. Because of these issues and the availability of easier to use circuits, notch filters are not discussed in this tutorial. Another advantage of this configuration is that the ratio of the largest resistor value to the smallest resistor value, and the ratio of the largest capacitor value to the smallest capacitor value (component spread) are low, which is beneficial for manufacturability. The frequency and Q terms are somewhat independent, but they are very sensitive to the gain parameter. The Sallen-Key is very Q-sensitive to element values, especially for high Q sections. The design equations for the Sallen-Key low-pass filter are shown in the Sallen-Key Low-Pass Design Equations section. Rev. A | Page 1 of 3 OUT R2 IN R1 C1 C2 R3 R4 R5 Figure 3. Sallen-Key Band-Pass Filter 10427-003 C2 MT-222 Mini Tutorial SALLEN-KEY LOW-PASS DESIGN EQUATIONS SALLEN-KEY HIGH-PASS DESIGN EQUATIONS + H ω0 2 + H s2 IN s 2 + α ω0 s + ω0 2 C1 R1 C1 OUT R2 R2 R3 R3 10427-004 R4 R4 Figure 4. H s 2 OUT C2 C2 VO = VIN R1 IN Figure 5. 1 R1 R 2 C1 C 2 1 1 1 (1 − H ) + s + + 1 2 C1 R 2 C 2 R R 10427-005 s 2 + α ω0 s + ω0 2 H s2 VO = VIN 1 + 2 1 R R C1 C 2 s 2 C2 C 2 C1 R 2 + R 2 + (1− H ) R1 1 +s + C 1 C 2 R 1 R 2 C1 C 2 To design the filter, choose C1 and R3. To design the filter, choose C1 and R3. Then Then k = 2 π F0 C1 k = 2 π F0 C1 R4 = m= C2 = C1 R3 (H − 1) α2 4 R1 = + (H − 1) R2 = C2 = m C1 2 R1 = αk R2 = α 2 mk Rev. A | Page 2 of 3 α + α 2 + (H − 1) 4k 4 α + α + (H − 1) 2 + 1 k Mini Tutorial MT-222 SALLEN-KEY BAND-PASS DESIGN EQUATIONS + H ω0 s s + α ω0 s + ω0 2 2 OUT R2 R1 C1 C2 R3 R4 R5 10427-006 IN Figure 6. VO = VIN 1 R1 C 2 C1 (C1 + C 2) C 2 C1 (1 − H ) + + R3 + R1 + R 2 1 2 R R R 1 2 2 s + s + C1 C 2 R3 + C1 C 2 R1 R 2 Hs To design the filter, choose C1 and R4. Then k = 2 π F0 C1 R5 = R4 H −1 C2 = 1 C1 2 R1 = 2 k R2 = 2 3k R3 = 4 k H= 1 1 6.5 − 3 Q REFERENCES Sallen, R. P. and E. L. Key, 1955. “A Practical Method of Designing RC Active Filters.” IRE Transactions on Circuit Theory, Vol. CT-2, 74–85. Zumbahlen, Hank, editor, 2008. Linear Circuit Design Handbook, Newnes, ISBN 978-0-7506-8703-4. REVISION HISTORY 7/12—Rev. 0 to Rev. A Changes to Statements following Equations .................................. 2 3/12—Revision 0: Initial Version ©2012 Analog Devices, Inc. All rights reserved. Trademarks and registered trademarks are the property of their respective owners. MT10427-0-7/12(A) Rev. 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