Prof. D. Manstretta LEZIONI DI FILTRI ANALOGICI. Danilo Manstretta AA

Transcription

1 AA-3 LEZIONI DI FILTI ANALOGICI Danilo Manstretta AA -3

2 AA-3 High Order OA-C Filters H() s a s... a s a s a n s b s b s b s b n n n n... The goal of this lecture is to learn how to design high order OA-C filters using real OTAs.

3 AA-3 3 Ideal Operational Amplifier symbol equivalent circuit e - e e + i o e e + i A(e + -e - ) e - In an ideal op-amp we assume: input resistance i approaches infinity, thus i = output resistance o approaches zero amplifier gain A approaches infinity

4 AA-3 4 Operational Transconductance Amplifier symbol equivalent circuit i e - e e + e e + Gm(e + -e - ) o C o e - In a real Operational Transconductance Amplifier (OTA) we assume: input resistance i approaches infinity, thus i = Finite DC gain (A =Gm o ) Finite gain-bandwidth product (Gm/C ) Gm has a finite value

5 AA-3 5 Cacaded Biquads Implementation H(s) (s s z,)(s s * z, ) (s s p, )(s s * p, ) (s s z,)(s s * z, ) (s s p, )(s s * p, ) (s s z,3)(s s * z,3 ) (s s p,3 )(s s * p,3 ) B B B3 s or /s biquads

6 AA-3 SINGLE-OTA BIQUADS Sallen-Key, auch

7 AA-3 7 Low-Pass auch Filter Q C C C C H(s) = + sc s C C 3 G

8 AA-3 8 auch Low-Pass with eal OTA In order to simplify the equations, consider a specific implementation: C =4C =4C = = 3 = Ideal transfer function: What is the effect of the real OTA on the transfer function? H(s) = + s4c + s 8 C ω = C Q=/ G = -

9 AA-3 9 auch Low-Pass with eal OTA (ii)

10 AA-3 auch Low-Pass with eal OTA (iii) Zero is located in the right half-plane (HP) zero close to g m /C Complex poles frequency is shifted and Q is modified: limited gm and GBW lead to poles Q enhancement, finite gain leads to Q degradation Additional (real) pole: around the OTA unity gain frequency (- g m /C )

11 AA-3 Sallen-Key Biquad (ii) C V IN C C C a b Q C C C C C H(s) = C C + s μ C C + + μ + s C C C G +s /Q+s

12 AA-3 Sallen-Key Design Strategies Five design elements, two main properties (G is less important): several design degrees of freedom Q G C C C C C C C Design. = =, C =C =C =/(C) G=3-/Q Note: Q is independent of,c Design. G= ( a = b ), C =C =C =/( C ) =Q/ C Note: / =Q Design 3. G=, = = =/( C C ) C =Q/ C ; C =/Q C Note: C /C =4Q

13 AA-3 3 Sallen-Key: finite op-amp gain The inverting and non-inverting terminals are not virtually shorted E =A (E 4 -V - ) E E E E 3 4 C C b a E 4 E E a b a E A E 4 μ = + B A A E / E 4 μ = μ + μ A

14 AA-3 4 Sallen-Key with OTA In order to simplify the equations, consider a specific implementation: C =C =C = = = H(s) = + Cs + C s C E E E E 3 4 C ω = C Q=/ G=

15 AA-3 5 Sallen-Key with real op-amp (OTA) a = C g m a = C g m b = + A b = C 4b + C + C g m + a s + a s H(s) = b + b s + b s + b 3 s 3 zeros and 3 poles b = b C + 4C C + C g m b 3 = C C g m

16 AA-3 6 Sallen-Key with real op-amp (OTA) (ii) The complex zeros have Q=(g m ) ½ / >> zeros are close to being pure imaginary: notch in H(s) at the zero frequency! The frequency of the zeros is Assuming the OTA DC gain g m >> and isolating the terms corresponding to the ideal transfer function from the parasitic pole, the denominator can be approximated as: g m C + sc + g m + s C + g m + sc g m The complex poles change in frequency and Q A parasitic real pole has appeared around gm/c

17 AA-3 7 TWO-OTA BIQUADS Fleisher-Tow, KHN, Tow-Thomas

18 AA-3 Fleischer-Tow Biquad G 6 C C 3 E E s s C C C s s C C C Q 3 C C

19 AA-3 Kerwin-Huelsman-Newcomb (KHN) C C C C Q G C E 5 C 3 E 4 6 G E E s s Q

20 AA-3 Two-Integrators Biquad Design -b b a E S S -/s -/s E b Q P a b E a E s b s b E a C b C E b

21 AA-3 Tow-Thomas 3 E 4 C C E E s C 4 s C C 3 E E E s G s Q G 3 4 Q C 3 3

22 AA-3 Filter Design Using eal OPAMP What happens if we replace the ideal OPAMP with a real circuit? Let s consider the impact of circuit limitations on the active integrator, the basic building block of high order filters designs.

23 AA-3 3 OTA-C Integrator C V IN VO Ideal integrator: v v IN sc g m (V - -V + ) O C O What happens if the ideal OA is replaced with a real OTA? Compute the transfer function as a function of the OTA parameters Derive design guidelines for the OTA

24 AA-3 4 OTA-C Integrator Analysis To keep equations simple lets consider different effects separately: Finite DC gain Finite gm Output capacitance Study effect on poles Q: define Integrator quality factor (Q INT )

25 AA-3 5 Finite DC Gain Neglect output/load capacitance (C =), G m is very large but the DC gain (G m ) is finite and equal to A. v IN i IN C e + v Gm(e + -e - ) o v vo sc Gm scv O sc v v G O m A G A vin sc vo v O m O sc O vin v v iin v vo sc Gmv O O vo A v sc A IN

26 AA-3 6 Finite DC Gain () AMPLITUDE vo A v sc A IN PHASE The effect of a finite OTA DC gain is to move the pole from the origin to = /(+A ).

27 AA-3 7 Finite DC Gain (3) vo A v sc A IN v v O IN 8 arctan A The effect of a finite OTA DC gain is to move the pole from the origin to = /(+A ). The phase shift at is slightly larger than 9 (phase lead)

28 AA-3 8 Finite Gm Neglect output/load capacitance (C =) and conductance: DC gain is infinite but G m has a finite value. C vosc Gm sc v v IN i IN e + v Gm(e + -e - ) v v IN O v v G G sc m m vin v iin v vo sc Gmv v sc G O v sc G IN m m

29 AA-3 9 Finite Gm () AMPLITUDE v sc G O v sc G IN m m PHASE z = The effect of a finite G m is to introduce a HP zero at G m /C and to move the unity gain frequency from /C to /(+/G m )C.

30 AA-3 3 Finite Gm (3) z = v s O z v sc G IN m v v O IN 9 arctan z z = G m /C The zero must be at a much higher frequency than. The phase shift at is slightly smaller than 9 (phase lag)

31 AA-3 3 Finite Gain-Bandwidth Product (GBW) Now consider a finite output/load capacitance C and a finite G m (GBW= G m C ). Neglect output conductance (infinite DC gain). TOT v IN i IN C e + v Gm(e + -e - ) C o vosctot Gm sc v vin sc vo v C C C sc v IN CTOT sc vo sc Gm sc sc O v v i v v sc G v v sc IN IN O m O O vo sc / Gm v sc sc / G IN O m C / C G O m

32 AA-3 3 Finite GBW () AMPLITUDE vo sc / Gm v sc sc / G IN O m PHASE z = p = 5 The effect of a finite GBW is to introduce an additional pole at approximately -G m /C O. The unity gain frequency is also slightly increased.

33 AA-3 33 Finite GBW (3) z = p = 5 vo s/ z v sc s / IN p v v O 9 arctan arctan IN z p z = G m /C p = G m /C O The phase shift at is slightly smaller than 9 (phase lag). The pole and the HP zero both contribute a phase lag

34 AA-3 34 Integrator Quality Factor (Q INT ) The cumulative effect of integrator non-idealities on the poles can be summarized in a single number: Q INT Q INT is a measure of the integrator phase deviations from 9 at the unity gain frequency. DEFINITION H j QINT tan tan 9 H. Khorramabadi and P.. Gray, High-frequency CMOS continuous time filters, JSSC Dec 984.

35 AA-3 35 Integrator Quality Factor: Q INT Ideal integrator eal integrator H j j H j jx H j H j e j arctan X Q INT Q INT X tan tan 9 This alternative definition may be easier to remember.

36 AA-3 36 Summary Finite DC Gain vo A v sc A IN Q INT X A Finite Gm (HP zero z =G m /C) v s O z v sc G IN m X / / z Q INT z Finite GBW (additional pole) vo v s s / IN p X p Q INT / p

37 AA-3 37 Summary () Q INT A z p Finite DC Gain gives a positive Q. ight half-plane zero (such as introduced by the finite Gm) and additional poles (such as introduced by load capacitance) introduce a negative Q. The two effects partially cancel each other but ensuring complete cancellation across process variations is not trivial. Design guidelines: large DC gain, ensure that zeros and poles are well above the poles frequency

38 AA-3 38 Biquad with Finite Integrator Q If a biquad is realized using integrators having finite Q INT, the poles Q P will be modified as follows: Q INT > Q Q INT < P Im(s) Q' P Q P Q INT Im(s) e(s) e(s) Q INT must be much higher than the poles Q to preserve the filter shape

39 AA-3 39 Biquad with Finite Integrator Q () If a biquad is realized using integrators having finite Q INT, the biquad transfer function will be modified as follows: H ' LP H If the error on the gain of the biquad at the pole frequency is to be lower than E, a specification on the minimum Q INT is derived: Q P Q INT LP Notice that the above considerations apply to any pair of poles of the filter transfer function, independent of the filter implementation (i.e. even if the filter is implemented as a ladder) Q P Q INT E db, W.J.A. De Heij, E. Seevinck, and K. Hoen, Practical Formulation of the elation Between Filter Specifications and the equirements for Integrator Circuits, TCAS Aug 989.

40 AA-3 4 Two-Integrators Biquad Design -b b a E S S -/s -/s E b Q P a b b C C E a E s b s b E a b E

41 AA-3 4 Integrator Non-Idealitities The effect of integrator non-idealities will be evaluated using the modified integrator transfer function H INT (s) -b a b b b Q P a E S S H INT (s) H INT (s) E H s a H s INT INT INT b H s b H s H INT (s) Finite DC Gain HP zero s s A s z Finite GBW (additional pole) s s/ p

42 AA-3 4 Finite DC Gain H INT / A s/ b Q P b H s a H s INT INT INT b H s b H s H s s s Q P A Q P A A ' Q P A A Q P A Assuming A >> Q ' P QP QP QP Q A Q P INT QINT A

43 AA-3 43 HP Zero H INT s s s z b Q P b ' H s a H s INT INT Q INT b H s b H s Pz z Q P z Assuming z >> H s s s / z s QPz z QP z Q ' P QP QP z Q Q Q P INT Q P z INT

44 AA-3 44 Finite GBW H INT s s s p b Q P b H s a H s INT INT INT b H s b H s H s s p s s Q ' P / Q ' P QP QP Q Q Q P p P INT Assuming p >> Q INT p

45 AA-3 45 Summary -b b a E S S H INT (s) H INT (s) E The effect of integrator non-idealities on the poles Q is: Finite DC Gain H INT QINT A / A s/ Q ' P QP Q Q P INT Finite Gm (HP zero) H INT s s s z Q INT z Finite GBW (additional pole) H s INT s s p Q INT p

46 AA-3 46 HIGHE ODE OPAMP-C FILTES Canonic Synthesis

47 AA-3 47 Multiple-Loop Feedback Architectures As the filter order increases, the Q of the poles increases and biquad based implementations become too sensitive to components variations and mismatches. Multiple-loop feedback architectures are to be preferred due to lower sensitivity The state-variable synthesis method allows the synthesis of an nth order filter (low-pass, band-pass or high-pass) in the form E a s... a s a s a n n n n n n... E s b s b s b s b

48 AA-3 48 State-Variable Method: All-pole filters E a E s b s b s b s b n n n n... E E n s b s... b s b s b E a E n n n s b s... b s b s b E E n n 3 n s b s... b s b s E b E E n n 3 E a E 3 n s... b s b E n s b s... b s b s E E b E n n 3 -b s s b s... b s b E E n n n 4 E 3 S E 4 n s b s E

49 AA-3 49 Controller Canonic Form ealization Follow-the-leader feedback architecture -b -b -b n- -b n- E a s n E s s n- E s s n- E se s E E a E s b s b s b s b n n n n... K. Laker, M. Ghausi, "Synthesis of a low-sensitivity multiloop feedback active C filter," IEEE Transactions on Circuits and Systems, Mar 974

50 AA-3 Inverting Integrators ealization -b n- (+/-) b b n- (+/-) b E a (-s) n E s (-s) n- E s (-s) n- E -se s E E a E s b s b s b s b n n n n...

51 AA-3 5 Inverse Follow-the-Leader Feedback

52 AA-3 Circuit ealization Example - b b b 3 b E a s s s s E E E s s s s

53 AA-3 Circuit ealization Example (ii) - b 3 b b b The feedback resistors are inversely proportional to the corresponding coefficient: i=/bi E a s s s s E E E s s s s

54 AA-3 General ealization E a s... a s a s a E s b s b s b s b n n n n n n... ealization of a generic transfer function with number of zeros strictly lower than the number of poles The realization with equal number of poles and zeros is only slightly more complicated

55 AA-3 55 Filter Design Steps Design specification FILTE MASK Normalization Approximation H A IN-BAND IPPLE STOP-BAND Network Function PASS-BAND Min ATTEN. ealization p s Denormalizati on

56 AA-3 56 Example 3 Design a 5th order all-pole low-pass filter using follow-theleader feedback architecture DC gain = db Chebyshev, 5th order, e=.59, =MHz VOUT a V s b s b s b s b s b IN

57 AA-3 57 Design Example of a Chebyshev Filter 5 th order, e=.588 v n sinh e n = (/5) sinh - (/.588) =.856 Sinh n =.895 ; Cosh n=.4 Normalized poles: k= s = sin(/) x.895 =.895; = cos(/) x.4 =.99 P = (s + ) / =.994; Q = P /(s ) = k= s = sin(3/) x.895 =.34; = cos(3/) x.4 =.6 P = (s + ) / =.655; Q = P /(s ) =.4 k=3 s = sin(5/) x.895 =.895; = cos(5/) x.4 = P3 =.894 u (k ) n (k ) (k ) s k s k j k sin n sinh v j cos n cosh v s s D s s s s P P () P P P3 QP QP

58 AA-3 58 Design Example of a Chebyshev Filter s s D s s s s P P () P P P3 QP QP Calculate the normalized polynomial coefficients: D() s s b s b s b s b s b b b Q Q 4 P3 P P P P P P P3 P P3 P 3 P P QPQP QP QP b P P P P P3 P P QP QP b P3 P P b P P P P P P P3 P P QPQP QP QP

59 AA-3 59 Flowgraph - b 3 b b 4 b E a s s s s s E b To de-normalize the coefficients: b k = b k 5-k If we substitute the integrators with /(sc), the feedback coefficients are changed as follows: b k =b k (C ) 5-k

60 AA-3 6 Circuit Implementation b4 b3 b b b C C C C C b V IN V OUT C b'' k b k bk b k

61 AA-3 6 BACKUP SLIDES MATEIALE INTEGATIVO

62 AA-3 6 Pole Quality Factor S Im(s) P The solutions of D(s)= (i.e. the poles of the network function) are complex numbers, typically represented in the S plane. s p = s p +j p S s P Q = ω σ p e(s) Solutions s k can be real or complex. Complex solutions always appear in conjugate pairs. A very important design parameter is the pole quality factor Q. The pole quality factor is given by the ratio between the pole frequency (distance from the origin in the s-plane) and the real part (distance from the imaginary axis in the s- plane)

63 AA-3 63 Biquad Magnitude esponse Two conjugate poles at H s = + s Qω + s ω Frequency response at H(j )=-jq Notice how the network function has been written, with explicit reference to and Q. This is usually done in biquad-based filter design since it greatly simplifies the design procedure.

64 AA-3 64 Biquad Phase esponse H s = + s Qω + s ω φ = arctan ωω Q ω ω f(j )=-/ Notice how, in low Q poles, the phase starts to move about a decade before the pole (i.e. much earlier than the magnitude response. As we shall see, low Q poles are desirable to lower power consumption and to minimize the sensitivity of the filter reponse to parameter variations.

65 AA-3 65 Appendix material (use only if needed) Poles and poles Q Multi-loop feedback architectures (summary) Ladder filter synthesis (summary) Frequency normalization and de-normalization

66 AA-3 66 Example Design of a low-pass biquad: DC gain= =MHz Q P =.7 E a E s b s b a b b Q P

67 AA-3 67 Flowgraph Implementation b b a E - S S -/s -/s E Modify flowgraph using flowgraph rules to simplify circuit implementation: divide and multiply by C. b C a C E S S E sc E sc 3 E 4 b C

68 AA-3 68 Flowgraph () To get a : equivalence with the circuit, let: - C /Q P E S S E sc E sc 3 E 4 Circuit implementation: C Q P C bk b k E E 3 E E E

69 AA-3 69 Generic Biquad Implementation E a s a s a E s b s b E E s b s b ' E E se ' 3 s E ' 6 Low-pass output Band-pass output High-pass output E 6 a a -a + E E a s a s a '

70 AA-3 7 Example Design of a low-pass biquad with transmission zero DC gain= =MHz Q=.7 z =MHz E a s a E s b s b a b b Q P a z

71 AA-3 7 Flowgraph E a a s E s b s b 5 a a + E

72 AA-3 7 Example Start with the poles. Modify flowgraph using flowgraph rules to simplify circuit implementation: divide and multiply by C b C a C E S S E sc E sc 3 E 4 b C E a E s b s b a b b Q P

73 AA-3 73 Example If we let C the flowgraph is simply: - /Q P E S S E sc E sc 3 E 4

74 AA-3 74 Example 3 Adder circuit implementation: E E E 4 E Q P b E E E 4 b E4 G E3 b G b E G E 4 b G Q P b G b E 3 Q b P G With this implementation we cannot set the gain independent of the poles Q! E E 4 4 Q P

75 AA-3 75 Zeros Implementation The zeros are now implemented with another adder: V E E OUT 4 z - /Q P E S S E sc E sc 3 E 4 z V OUT

76 AA-3 76 Circuit Implementation V E E OUT 4 z E V OUT z a E 4 a

77 AA-3 77 Final Circuit Implementation C C E E 4 E E 3 Q P z V OUT