Active Filters: concepts All input signals are composed of sinusoidal components of various frequencies, amplitudes and phases. If we are interested in a certain range of frequencies, we can design filters to eliminate frequency components outside the range Filters are usually categorized into four types: low-pass filter, high- pass filter, band-pass filter and band-reject filter. Low-pass filter passes components with frequencies from DC up to its cutoff frequency and rejects components above the cutoff frequency. Low-pass filter composed of OpAmp are called active filter (as opposed to lumped passive filter with resistor, capacitor and inductor) Active filters are desired to have the following characteristics:  Contain few components  Insensitive to component variation  Not-too-hard-to-meet specifications on OpAmp  Easy reconfiguration to support different requirements (like cutoff freq)  Require a small spread of component values

Applications of Analog Filters Analog filters can be found in almost every electronic circuit. Audio systems use them for pre-amplification, equalization, and tone control. In communication systems, filters are used for tuning in specific frequencies and eliminating others (for example, to filter out noise). Digital signal processing systems use filters to prevent the aliasing of out- of-band noise and interference.

Butterworth low-pass filter Many low-pass filter are designed to have a Butterworth transfer function with magnitude response as follows: Graphs from Prentice Hall

Low-pass filter: Sallen-Key Circuits Active low-pass Butterworth filter can be implemented by cascading modified Sallen-Key circuits. The Sallen-Key circuit itself is a 2 nd order filter. To obtain an nth order filter, n/2 SK circuits should be cascaded During design, capacitance can be selected first and then resistor values. As K increase from 0 to 3, the transfer function displays more and more peaking. It turns out that if K>3, then the circuit is not stable. Empirical values have been found for filters of different orders

Example of a 4 th -order Lowpass filter by cascading two 2 nd -order SK filters

Comparison of gain versus frequency for the stages of the fourth-order Butterworth low-pass filter.

Butterworth high-pass filter By a change, the lowpass Butterworth transfer function can be transformed to a high-pass function.

Butterworth high-pass filter: Sallen-Key By a change, the lowpass Butterworth transfer function can be transformed to a high-pass function. With real OpAmp, the Sallen-Key is not truly a high-pass filter, because the gain of the OpAmp eventually falls off. However, the frequencies at which the OpAmp gain is fairly high, the circuit behaves as a high-pass filter. Since the high-pass Sallen Key circuit is equivalent the same as the low-pass one, the empirical values for K would be still valid in this case also.

Band-pass filter: Sallen-Key Circuits Graphs from Prentice Hall If we need to design a band-pass filter in which the lower cutoff frequency is much less than the upper cutoff frequency, we can cascade a low-pass filter with a high-pass filter. The below band-pass filter uses the first stage as a low-pass filter which passes frequency less than 10KHz and the second stage as a high-pass filter that passes only frequency above 100Hz. Thus, frequency components in-between is passed to the output.

Figure Bode plots of gain magnitude for the active filter of Example 11.2.

A summary

Application: Equalizer (EQ)