Sallen and Key Two Pole Filter Buffer amplifier

But Apply Kirchoff’s current law to v 1 node:

Substituting,

Two-Pole Low-Pass Filter

Two-Pole High-Pass Filter

Two-Pole Example Design a second order normalised Butterworth low pass filter. So, Eg.

Higher Order Filters Using a single op-amp the highest order practical filter design is two. For higher order filters, more than one op- amp is needed. Simplest design technique is cascade synthesis or synthesis by sections.

Synthesis by Sections When two (or more) filters are cascaded in series, the total transfer function is the product of the individual transfer functions. H TOT (s) = H 1 (s) H 2 (s) The required transfer function must be factorised into second order sections. H1(s)H1(s)H2(s)H2(s) V IN (s)V IN (s)H 1 (s)V IN (s)H 1 (s) H 2 (s)

Factorising the Transfer Function Filter design techniques usually generate the positions of the poles (and zeros) of the transfer function. Grouping complex conjugate pairs together is essential for physically realisable sections. If the required filter order is odd, a single pole will be left over. This is realised by a first order section.

Fourth Order Example The poles of a normalised fourth order low pass Butterworth filter are: Transfer function is, therefore:

Fourth Order Realisation

Third Order Example The poles of a normalised third order low pass Bessel filter are: Transfer function is, therefore:

Ordering the Sections In theory, using ideal op-amps, it doesn’t matter which section is first or second. In practice, the decision is based on considerations of: Saturation Levels Noise Figure From fourth order example:

Second Order Frequency Response Frequency [rad/s] Gain  =2  =  If the damping ratio, , is less than 1/  2, the peak gain is greater than one.  This means that the maximum input level is reduced, to avoid op-amp saturation.

Butterworth Filter Example

At  = 1 rad/s, H 1 (s)H 2 (s) H 1 (s)

Section Ordering To get the highest maximum input signal range. Sections with the largest damping ratio (lowest Q) should come first. Resonant sections with low damping ratios come last, possibly susceptible to noise. To get the lowest output noise. Highest gain sections (i.e. most resonant) should come first. Subsequent sections attenuate noise. What’s the best order ? It depends.

Summary The Sallen and Key configuration realises a two-pole filter section using a single op-amp. Synthesis by sections creates higher order filters by cascading first and second order sections. Under ideal assumptions, the order of the sections is irrelevant. Practically, dynamic range considerations decide the order of the sections.