1. Analog Filters: Biquad Circuits Franco Maloberti

2. Analog Filters: Biquad Circuits2 Introduction Active filters which realize the biquadratic transfer function are important building blocks (biquad) pp pp 00

3. Franco MalobertiAnalog Filters: Sensitivity3 Introduction Biquads can build high-order filters Poles and zeros are Real or complex conjugate s or 1/s B1B2B3 Problem: how to properly pair poles and zeros

4. Franco MalobertiAnalog Filters: Sensitivity4 Single Amplifier Configurations RC A + - A + - R R(k-1) RC A + - R R(k-1) Enhanced positive or negative feedback

5. Franco MalobertiAnalog Filters: Sensitivity5 Sallen-Key Biquad R1R1 R2R2 C1C1 C2C2 E1E1 E2E2 Only real poles (or zeros) C1C1 C2C2 E1E1 E2E2  The feedback permits us to achieve complex poles

6. Franco MalobertiAnalog Filters: Sensitivity6 Sallen-Key Biquad (ii) C1C1 C2C2 E1E1  E2E2 R1R1 R2R2 RbRb RaRa E3E3 E4E4

7. Franco MalobertiAnalog Filters: Sensitivity7 Sallen-Key Biquad (ii) Five design elements, two properties (G is not important) Case 1: C 1 =C 2 ; R 1 =R 2 =R R=1/  0  =3-1/Q Case 2: C 1 =C 2 ; R a =R b R 1 =Q/  0 R 2 =1/Q  0 Case 3: R 1 =R 2 ;  =1 C 1 =2Q/  0 C 2 =1/2Q  0 Case 4: C 1 =3 1/2 Q C 2 ;  =4/3 R 1 =1/Q  0 R 2 =1/3 1/2  0

8. Franco MalobertiAnalog Filters: Sensitivity8 Sallen-Key Biquad (iii) Sensitivities

9. Franco MalobertiAnalog Filters: Sensitivity9 Sallen-Key High- and Band-pass R1R1 R2R2 C1C1 C2C2 E1E1 E2E2 R1R1 R2R2 C1C1 C2C2 E1E1 E2E2 R1R1 R2R2 C1C1 C2C2 E1E1 E2E2 C1C1 C2C2  LP HP BP

10. Franco MalobertiAnalog Filters: Sensitivity10 Generic Sallen-Key

11. Franco MalobertiAnalog Filters: Sensitivity11 Sallen-Key: finite op-amp gain The inverting and non-inverting terminals are not virtually shorted C1C1 C2C2 E1E1  E2E2 R1R1 R2R2 RbRb RaRa E3E3 E4E4

12. Franco MalobertiAnalog Filters: Sensitivity12 Sallen-Key in IC C1C1 C2C2 E1E1  E2E2 R1R1 R2R2 RbRb RaRa E3E3 E4E4 C1C1 C2C2 E1E1  E2E2 R1R1 R2R2 E3E3 E4E4

13. Franco MalobertiAnalog Filters: Sensitivity13 LP Sallen-Key with real op-amp

14. Franco MalobertiAnalog Filters: Sensitivity14 LP Sallen-Key with real op-amp (ii) The transfer function has two zeros and three poles. If k = Rgm >> 1 the zeros are practically complex conjugates and are located at The extra-pole is real and is located around the GBW of the op-amp.

15. Franco MalobertiAnalog Filters: Sensitivity15 LP Sallen-Key with real op-amp (iii) Possible responses

16. Franco MalobertiAnalog Filters: Sensitivity16 Sallen-Key IC Implementations

17. Franco MalobertiAnalog Filters: Sensitivity17 Band-reject Biquad A band-reject response requires zeros on the immaginary axis It can be obtained with the generic SK implementation Another option is to use a twin-T network

18. Franco MalobertiAnalog Filters: Sensitivity18 Band-reject Biquad (ii) Using complementary values

19. Franco MalobertiAnalog Filters: Sensitivity19 Use of Feed-forward Assume High-passBand-pass

20. Franco MalobertiAnalog Filters: Sensitivity20 Infinite-Gain Feedback Biquad Sallen-Key architectures require input common mode range. Input parasitic capacitance of the op-amp can affect the filter response Keep the inputs of the op-amp at ground or virtual ground

21. Franco MalobertiAnalog Filters: Sensitivity21 Infinite-Gain Multi-Feedback Biquad A conventional op-amp amplifier is not able to realize complex-conjugate poles Two or more feedback connections achieve the result

22. Franco MalobertiAnalog Filters: Sensitivity22 Low-Pass MFB

23. Franco MalobertiAnalog Filters: Sensitivity23 Design and Sensitivity Five elements and three equations “Arbitrarily choose two of them and determine the remaining three parameters Assess the “quality of design” Sensitivity on relevant design element Spread of components Cost of the implementation Linearity of components

24. Franco MalobertiAnalog Filters: Sensitivity24 High-pass and Band-pass

25. Franco MalobertiAnalog Filters: Sensitivity25 Two-Integrators Biquad Use of state-variable method Derive the block diagram Translate the block diagram into an active implementation Addition or subtraction Integration Dumped integration (integration plus addition)

26. Franco MalobertiAnalog Filters: Sensitivity26 Basic Blocks

27. Franco MalobertiAnalog Filters: Sensitivity27 State Variables The state variable are relevant voltages of the network E1E1 E2E2 E1E1 E2E2 E5E5 Ga0a0 E6E6 E5E5 E6E6 E4E4

28. Franco MalobertiAnalog Filters: Sensitivity28 State Variables (ii) E4E4 E3E3 E3E3 E6E6

29. Franco MalobertiAnalog Filters: Sensitivity29 State Variables (iii) E3E3 E6E6 E4E4 E3E3

30. Franco MalobertiAnalog Filters: Sensitivity30 State Variables (iv) + a2a2 -a 1 a0a0 E2E2

31. Franco MalobertiAnalog Filters: Sensitivity31 Implementations Kervin-Huelsman-Newcomb Tow-Thomson Fleischer-Tow …. Fleischer-Laker

32. Franco MalobertiAnalog Filters: Sensitivity32 Implementations (ii)