Analog Electronics Active Filters Lecture 19

AE

1. Analog Electronics Lecture 19 : Active Filters contd … Mon 26 Feb 2024 Department of Electrical and Electronics Engineering BITS Pilani Hyderabad Campus

2. In case of LPF from the given specifications of p, ω s, A max (dB), A min (dB) the ϵ and N (order of the filter) can be determined .  

3. Standard prototype for filter design is Butterworth. In case of LPF from the given specifications of p, ω s, A max (dB), A min (dB) the ϵ and N (order of the filter) can be determined .  

4. To achieve a low-pass Butterworth response, we need to create a transfer function whose poles are arranged as follows: For a Butterworth filter the poles will lie on a circle of radius = and are separated from each other by an angle of π /N and the first and last pole are separated from the imaginary axis by an angle of π /2N.  

5. Poles have  equal angular  spacing and lie along a  semicircular path  in the left half-plane. Distance between the origin and each pole is the same, and this in turn means that all poles  have the same frequency ω . Angle that separates the poles is equal to 180°/N, where N is the order of the filter. In the example above, N = 4, and the separation angle is 180°/4 = 45°. First and last pole are separated from the imaginary axis by an angle of π /2N. Equal angular spacing of the Butterworth poles indicates that even-order filters will have only complex-conjugate poles. Odd-order filters have complex-conjugate poles plus one purely real pole that lies along the negative real axis at a distance of ω from the origin. All poles have the same ω , but the horizontal distance from the origin varies. Thus, the poles  have different Q factors.

6. All pole filter: T(s) =  

7. Butterworth filter polynomials The polynomials (denominator of the transfer function) are obtained after normalization. The normalized Butterworth polynomial equations have the general form: N polynomial

8. Butterworth filter polynomials - roots

9. Filter realization First and second order filters are the simplest to design. For higher order filters first and second order filters are connected in cascade. Filters are realized using active RC circuits. First order LOW PASS FILTER

10. First order LOW PASS FILTER

11. First order LOW PASS FILTER Normalized transfer function T(j ω ) = K/(s n +1) where s n = j ω / ω o

12. First order LOW PASS FILTER

13. First order HIGH PASS FILTER ω o = 1/ RC T(s) = Ks/ ω o / [1+ (s/ ω o )] T(s) = Ks / [s + ω o ] Normalized transfer function T(j ω ) = Ks n /(s n +1) where s n = j ω / ω o

14. First order BAND PASS FILTER

15. First order BAND PASS FILTER

16. First order BAND PASS FILTER

17. First order BAND PASS FILTER

18. ALL PASS FILTER Passes all frequency components of an input signal without attenuation (hence ALL PASS ) but provides phase shift depending on frequency of the input signal ( PHASE SHIFTER ). Phase shift is equivalent to time delay hence time delay circuit. For an incremental shift in signal frequency a predictable change in time delay results as the signal passes through the filter. When signals are transmitted over transmission lines (telephone wires) signals undergo change in phase, these phase changes will be compensated by all pass filters. Hence called as Delay equalizers or Phase correctors .

19. ALL PASS FILTER V p

20. It has a zero at 1/RC and a pole at -1/RC Circuit introduces a variable phase shift (lag) from 0 to -180° with a value of - 90° at ω = ω o . Op amp with large bandwidth is preferred. ALL PASS FILTER

21. ALL PASS FILTER Interchange R and C in the circuit shown to get a phase lead circuit Phase response of that circuit will be 180° - 2 tan _1 (ω/ ω o )