1. EECE 352 Winter 2008Ch. 12 Active Filters1 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 (gain and filtering) 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)

2. EECE 352 Winter 2008Ch. 12 Active Filters2 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

3. EECE 352 Winter 2008Ch. 12 Active Filters3 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

4. EECE 352 Winter 2008Ch. 12 Active Filters4 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)

5. EECE 352 Winter 2008Ch. 12 Active Filters5 First Order Filter Functions * First order filter functions are of the general form Low Pass High Pass a 1 = 0 a 0 = 0

6. EECE 352 Winter 2008Ch. 12 Active Filters6 First Order Filter Functions * First order filter functions are of the form General All Pass a 1  0, a 2  0

7. EECE 352 Winter 2008Ch. 12 Active Filters7 Example of First Order Filter - Passive *Low Pass Filter 0 dB

8. EECE 352 Winter 2008Ch. 12 Active Filters8 Op Amp Characteristics *Consider only ideal op amp’s in our study of active filters. Note: Since the open-loop gain A is infinite, there needs to be virtually no voltage difference between the two inputs to get a finite output. Ex. For A = 100,000 and V out = 1 V, then v + – v - = V out / A = 1V/100,000 = 0.00001 V So for our analysis of op amps in active filters, we will frequently make the approximation that v + – v - ≈ 0 or simply v + ≈ v -.

9. EECE 352 Winter 2008Ch. 12 Active Filters9 20 log (R 2 /R 1 ) Example of First Order Filter - Active *Low Pass Filter V_= 0 IoIo I 1 = I o GainFilter function I_= 0

10. EECE 352 Winter 2008Ch. 12 Active Filters10 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.

11. EECE 352 Winter 2008Ch. 12 Active Filters11 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

12. EECE 352 Winter 2008Ch. 12 Active Filters12 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

13. EECE 352 Winter 2008Ch. 12 Active Filters13 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

14. EECE 352 Winter 2008Ch. 12 Active Filters14 *Low pass filter Example - Passive Second Order Filter Function General form of transfer function T(dB)  00 0 dB Q

15. EECE 352 Winter 2008Ch. 12 Active Filters15 Example - Passive Second Order Filter Function *Bandpass filter General form of transfer function T(dB)  00 0 dB -3 dB

16. EECE 352 Winter 2008Ch. 12 Active Filters16 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 ViVi VoVo ViVi VoVo

17. EECE 352 Winter 2008Ch. 12 Active Filters17 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 Note: C 3 = C 4 = C

18. EECE 352 Winter 2008Ch. 12 Active Filters18 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 ViVi VoVo