ECES 352 Winter 2007 Ch. 12 Active Filters Part 1 1 Active Filters *Based on use of amplifiers to achieve filter function *Frequently use op amps so filter may have some gain as well. *Alternative to LRC-based filters *Benefits Provide improved characteristics Smaller size and weight Monolithic integration in IC Implement without inductors Lower cost More reliable Less power dissipation *Price Added complexity More design effort Transfer Function V o (s) V i (s)

ECES 352 Winter 2007 Ch. 12 Active Filters Part 1 2 Filter Types *Four major filter types : Low pass (blocks high frequencies) High pass (blocks low frequencies) Bandpass (blocks high and low frequencies except in narrow band) Bandstop (blocks frequencies in a narrow band) Low PassHigh Pass BandpassBandstop

ECES 352 Winter 2007 Ch. 12 Active Filters Part 1 3 Filter Specifications *Specifications - four parameters needed Example – low pass filter: A min, A max, Passband, Stopband Parameters specify the basic characteristics of filter, e.g. low pass filtering Specify limitations to its ability to filter, e.g. nonuniform transmission in passband, incomplete blocking of frequencies in stopband

ECES 352 Winter 2007 Ch. 12 Active Filters Part 1 4 Filter Transfer Function *Any filter transfer function T(s) can be written as a ratio of two polynomials in “s” *Where M < N and N is called the “order” of the filter function Higher N means better filter performance Higher N also means more complex circuit implementation *Filter transfer function T(s) can be rewritten as where z’s are “zeros” and p’s are “poles” of T(s) where poles and zeroes can be real or complex *Form of transfer function is similar to low frequency function F L (s) seen previously for amplifiers where A(s) = A M F L (s)F H (s)

ECES 352 Winter 2007 Ch. 12 Active Filters Part 1 5 First Order Filter Functions * First order filter functions are of the general form Low Pass High Pass a 1 = 0 a 0 = 0

ECES 352 Winter 2007 Ch. 12 Active Filters Part 1 6 First Order Filter Functions * First order filter functions are of the form General All Pass a 1  0, a 2  0

ECES 352 Winter 2007 Ch. 12 Active Filters Part 1 7 Example of First Order Filter - Passive *Low Pass Filter 0 dB

ECES 352 Winter 2007 Ch. 12 Active Filters Part log (R 2 /R 1 ) Example of First Order Filter - Active *Low Pass Filter V_= 0 IoIo I 1 = I o GainFilter function

ECES 352 Winter 2007 Ch. 12 Active Filters Part 1 9 Second-Order Filter Functions * Second order filter functions are of the form which we can rewrite as where  o and Q determine the poles * There are seven second order filter types: Low pass, high pass, bandpass, notch, Low-pass notch, High-pass notch and All-pass  jj s-plane oo x x This looks like the expression for the new poles that we had for a feedback amplifier with two poles.

ECES 352 Winter 2007 Ch. 12 Active Filters Part 1 10 Second-Order Filter Functions Low Pass High Pass Bandpass a 1 = 0, a 2 = 0 a 0 = 0, a 1 = 0 a 0 = 0, a 2 = 0

ECES 352 Winter 2007 Ch. 12 Active Filters Part 1 11 Second-Order Filter Functions Notch Low Pass Notch High Pass Notch a 1 = 0, a o = ω o 2 a 1 = 0, a o > ω o 2 a 1 = 0, a o < ω o 2 All-Pass

ECES 352 Winter 2007 Ch. 12 Active Filters Part 1 12 Passive Second Order Filter Functions *Second order filter functions can be implemented with simple RLC circuits *General form is that of a voltage divider with a transfer function given by *Seven types of second order filters High pass Low pass Bandpass Notch at ω o General notch Low pass notch High pass notch

ECES 352 Winter 2007 Ch. 12 Active Filters Part 1 13 *Low pass filter Example - Passive Second Order Filter Function General form of transfer function T(dB)  00 0 dB Q

ECES 352 Winter 2007 Ch. 12 Active Filters Part 1 14 Example - Passive Second Order Filter Function *Bandpass filter General form of transfer function T(dB)  00 0 dB -3 dB

ECES 352 Winter 2007 Ch. 12 Active Filters Part 2 15 Butterworth Filters *Second order filters *Can be low or high pass. *Provide improved performance: No peak near band edge that is seen for other filters, i.e. it is maximally flat unlike other second order filters which give the shape shown below Falloff for Butterworth filter is steeper, i.e. 40 dB/dec rather than 20 db/dec for passive RLC filters. High Pass Filter Low Pass Filter

ECES 352 Winter 2007 Ch. 12 Active Filters Part 2 16 Low Pass Butterworth Filter General form for biquadratic filter This has form for a low pass biquadratic filter I C2 I C1 I R1 I R2 VoVo VoVo V 12 ViVi

ECES 352 Winter 2007 Ch. 12 Active Filters Part 2 17 Low Pass Butterworth Filter Design T(dB) 00 0 dB Q(dB) * Given the filter specification (  0 ), we can determine the R and C. * One specification, two parameters – R and C * Pick a convenient value, say C = 5 nF. * Calculate R from C and ω o. NOTE 40 dB/dec