1. 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 lProvide improved characteristics lSmaller size and weight lMonolithic integration in IC lImplement without inductors lLower cost lMore reliable lLess power dissipation *Price lAdded complexity lMore design effort Transfer Function V o (s) V i (s)

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

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

4. 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 lHigher N means better filter performance lHigher N also means more complex circuit implementation *Filter transfer function T(s) can be rewritten as lwhere z’s are “zeros” and p’s are “poles” of T(s) lwhere 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. 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

6. 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

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

8. ECES 352 Winter 2007 Ch. 12 Active Filters Part 1 8 20 log (R 2 /R 1 ) Example of First Order Filter - Active *Low Pass Filter V_= 0 IoIo I 1 = I o GainFilter function

9. 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.

10. 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

11. 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

12. 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 lHigh pass lLow pass lBandpass lNotch at ω o lGeneral notch lLow pass notch lHigh pass notch

13. 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

14. 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

15. ECES 352 Winter 2007 Ch. 12 Active Filters Part 1 15 Single-Amplifier Biquadratic Active Filters *Generate a filter with second order characteristics using amplifiers, R’s and C’s, but no inductors. *Use op amps since readily available and inexpensive *Use feedback amplifier configuration lWill allow us to achieve filter-like characteristics *Design feedback network of resistors and capacitors to get the desired frequency form for the filter, i.e. type of filter, e.g bandpass. *Determine sizes of R’s and C’s to get desired frequency characteristics (  0 and Q), e.g. center frequency and bandwidth. *Note: The frequency characteristics for the active filter will be independent of the op amp’s frequency characteristics. Example - Bandpass Filter General form of transfer function

16. ECES 352 Winter 2007 Ch. 12 Active Filters Part 1 16 Design of the Feedback Network *General form of the transfer function for feedback network is *Loop gain for feedback amplifier is *Gain with feedback for feedback amplifier is *Poles of feedback amplifier (filter) are found from setting Conclusion: Poles of the filter are the same as the zeros of the RC feedback network ! Design Approach: 1. Analyze RC feedback network to find expressions for zeros in terms R’s and C’s. 2. From desired  0 and Q for the filter, calculate R’s and C’s. 3. Determine where to inject input signal to get desired form of filter, e.g. bandpass.

17. ECES 352 Winter 2007 Ch. 12 Active Filters Part 1 17 Design of the Feedback Network *Bridged-T networks (2 R’s and 2C’s) can be used as feedback networks to implement several of the second order filter functions. *Need to analyze bridged-T network to get transfer function t(s) of the feedback network. We will show that *Zeros of this t(s) will give the pole frequencies for the active filter.. Bridged – T network General form of filter’s transfer function

18. ECES 352 Winter 2007 Ch. 12 Active Filters Part 1 18 Analysis of t(s) for Bridged-T Network VbVb VaVa I 3 = (V b -V a )/R 3 I 2 = I 3 I a = 0 I1I1 V 12 I4I4 Analysis for t(s) = V a / V b

19. ECES 352 Winter 2007 Ch. 12 Active Filters Part 1 19 Analysis of Bridged-T Network *Setting numerator of t(s) = 0 gives zeroes of t(s), which are also the poles of filter’s transfer function T(s) since *Where the general form of filter’s T(s) is *Then comparing the numerator of t(s) and the denominator of T(s),  o and Q are related to the R’s and C’s by *so *Given the desired filter characteristics specified by  o and Q, the R’s and C’s can now be calculated to build the filter. These have the same form – a quadratic !