Continuous-Time Active Filters

1. Continuous-Time Active Filters

2. Introduction Filtering: most common linear time-invariant (LTI) signal processing function – selecting the signal bandwidth of interest (in reality neither linear nor time-invariant)… Categories: continuous time (CT), discrete time (DT) analog filter, digital filter (will focus on CT analog filters for this course) Frequency domain:low-pass (LP), high-pass (HP), band-pass (BP)… Time domain: impulse response, FIR, IIR…

3. Simple RC Filter Continuous-time, 1st-order, one real pole, low-pass

4. Simple RC Filter Frequency response Impulse response

5. General Filter Specs • Ideal LPF: • Non-causal • Infinite complexity • More realistic: • Magnitude response • ωp, ωs, αp, and αs • Phase response

6. Example: 2nd-order VTF • Continuous-time • Where are the poles? (complex conjugate poles for maximum flatness) • Low-pass, high-pass, or band-pass?

7. Passive RLC Filter  • No active component (low power) • Inductors are bulky and expensive to realize in IC’s • Values of R, L, and C will not track each other

8. Active OP-RC Filter • It’s active • Inductor-less • Area efficient • Values of R’s and C’s and their time co.’s won’t track each other – may lead to RC time constant variations of as high as 20% • RC time constant enters the VTF in product form – can be tuned for accuracy Assuming ideal op amps, 

9. Continuous-Time Integrator Assuming ideal op amp,

10. Cascade Filter Design‒ Biquads

11. Cascade Filter Design For a real-coefficient H(s): 2nd-order 1st-order Biquad: The leading minus sign in Hbq(s) is only for convenience

12. Special Cases of Biquad

13. Q Factor of Poles

14. Signal Flow Graph (SFG) for Biquad  Note: the partition of the biquadratic VTF is not unique

15. OP-RC Implementation

16. Alternative SFG for Biquad Recast Hbq(s): 

17. Alternative OP-R-C Prototype

18. Cascade Filter Design For a real-coefficient H(s): 2nd-order 1st-order • Order of cascade determines the signal dynamic range • Optimized using engineering rule of thumb or thru simulation

19. Biquad Cascade Filter Design • Most flexible arrangement of cascade filter design • Allow independent, non-interacting control of (ω0, Q) for pole pairs • Easy design • Components need to be scaled for maximum DR and minimum component spread • Pass-band sensitivity to capacitance variation is finite → Ladder filter can achieve zero sensitivity

20. Scaling of Active Filter

21. Typical Active Multi-Stage Filters Initial component values may not be optimal...

22. Freq. Response of Internal Nodes • Internal signal swings need to be large to max SNR • But not too large such that op amps saturate (producing distortion)

23. DR Scaling of ith Integrator (Vi) • First find out the peak value of Vi(ω), mostly done with simulation • Then find out the ratio ki = Vi,peak/Vo,max • Multiply all capacitors connecting at Vi by ki: Fi → Fi*ki, Sij → Sij*ki, … • Divide all resistors connecting at Vi by ki: Fi → Fi*ki, Sij → Sij*ki, … • Repeat for all internal nodes…

24. After DR Scaling Max internal signal swings all line up to Vo,max

25. Scaling for Min. Component Spread • Find out the smallest cap/res connected to Xi – the summing node of Ai • determine the optimum scaling factor mi to minimize spread • Multiply all capacitors connected to Xi by mi: Fi → Fi*mi, Sji → Sji*mi, … • Divide all resistors connected to Xi by mi: Fi → Fi*mi, Sji → Sji*mi, … • Repeat for all integrators…

26. Scaling of Active Filter • DR and min spread scaling do not take op-amp loading into account – lots of work if individual op amps are sized to meet the settling time constraint • Upon the completion of scaling, simulation needs to be performed on the resulting filter to find out the overall SNR • If SNR is lower than the spec, capacitors and op amps need to be scaled up and resistors scaled down to meet the SNR spec (think about how integrated output noise behaves)

27. Ladder Filter Design

28. Motivation • Cascade filter design • Sensitive to component variations, especially high-Q poles • Ladder filter design • Achieves zero sensitivity to component variations • Discrete CT LC filters with very high-Q poles are built with ladder structures over the years

29. Ladder Filter Reactance two-port • Doubly terminated reactance two-port network • Delivers the optimum power matching in the passband • ∂|Vout|/∂Zi = 0 for all L’s and C’s → low sensitivity

30. State Space of Ladder Filter Pick –V1, -I2, V3 as the state variables for synthesis

31. Signal Flow Graph (SFG)

32. CT OP-RC Ladder Filter • Three free state variables → three op amps • A.k.a the leapfrog ladder structure

33. Transmission Zeros

34. Transmission Zeros

35. Modified SFG with Derivatives

36. OP-RC Ladder Filter w/ Derivatives • Derivative input paths implemented with capacitors

37. Other Active Filters

38. Tow-Thomas Biquad • Low sensitivity • Non-interactive • tuning property [1] P. E. Fleischer and J. Tow, "Design formulas for biquad active filters using three operational amplifiers,“ Proceedings of the IEEE, vol. 61, pp. 662-3, issue 5, 1973.

39. Design Equations for Tow-Thomas  Note: C1, C2, k1, k2, R8 are free parameters

40. Sallen-Key LPF • OP-RC active filters are ideally insensitive to bottom-plate stray caps • Sallen-Key is sensitive to bottom-plate parasitics at node A and B

41. Design Equations for SK LPF 

42. Sallen-Key BPF Still sensitive to parasitic capacitance at node A and B

43. Design Equations for SK BPF 

44. MOSFET-C Active Filter

45. MOSFET Resistor • MOSFET in triode region is a variable resistor • Compact, low parasitics compared to large-value resistors

46. MOSFET Resistor But the large-signal response is quite nonlinear

47. A Linear (Diff.) MOSFET Resistor MOSFET resistor is linear when driven by balanced differential signals!

48. Rudell VGA + Mixer • M9-M15 comprise • the CMFB circuit • Gain adjustment • by varying IGain [2] J. C. Rudell et al., “A 1.9-GHz wide-band IF double conversion CMOS receiver for cordless telephone applications,” IEEE Journal of Solid-State Circuits, vol. 32, pp. 2071-88, issue 12, 1997.

49. MOSFET-C Integrator = • Sources of M1 and M2 are ideally always equal-potential • Fully differential circuit rejects the 2nd-order harmonic (and all even-order distortions) • Triode resistance significantly depends on process (threshold, mobility, etc.), temperature, and VDD→ Filter response needs tuning

50. Frequency Tuning