EE247 Lecture 3. Signal flowgraph concept First order integrator based filter Second order integrator based filter & biquads

Transcription

1 Summary last week EE47 Lecture 3 Integrator based filters Signal flowgraph concept First order integrator based filter Second order integrator based filter & biquads High order & high Q filters Cascaded biquads Cascaded biquad sensitivity Ladder type filters EECS 47 Lecture 3: Filters 4 H.K. Page Summary Last Week Major success in CMOS technology scaling: Inexpensive DSPs technology esulted in the need for high performance Analog/Digital interface circuitry Analog/Digital interface building blocks includes Analog filters D/A converters A/D converters EECS 47 Lecture 3: Filters 4 H.K. Page

2 Summary Last Week Analog filters: Filter specifications Quality factor Frequency characteristics Group delay no phase distortion constant group delay Filter types Butterworth Maximally flat passband amplitude Chebyshev I ipple in the passband Chebyshev II ipple in the stopband Elliptic ipple in both passband & stopband highest rolloff/pole Bessel Maximally flat group delay Group delay comparison example phase distortion effects on BE EECS 47 Lecture 3: Filters 4 H.K. Page 3 Summary LPF OutofBand Signal Attenuation Versus Filter Order Example: Butterworth lowpass filter Magnitude rolloff versus filter order (n): Out of band magnitude rolloff nxdb/decade Magnitude (db) n= Normalized Frequency EECS 47 Lecture 3: Filters 4 H.K. Page 4

3 Summary Comparison of Various LPF Magnitude esponse Magnitude (db) 4 6 Normalized Frequency Magnitude (db) All 5th order filters with same corner freq. Bessel Butterworth Chebyshev I Chebyshev II Elliptic EECS 47 Lecture 3: Filters 4 H.K. Page 5 Summary Last Week Monolithic LC Filters Monolithic inductor in CMOS tech. Integrated L<nH with Q< combined with max. cap. pf LC filters in the monolithic form feasible: freq>5mhz Analog/Digital interface circuitry require fully integrated filters with critical frequencies << 5MHz Good alternative: Integrator based filters EECS 47 Lecture 3: Filters 4 H.K. Page 6

4 Start from C prototype Integrator Based Filters First Order LPF Use KCL & KVL to derive state space description: V I s I C Vo Vo = s C Use state space description to draw signal flowgraph (SFG) What is Signal Flowgraph (SFG)? EECS 47 Lecture 3: Filters 4 H.K. Page 7 What is a Signal Flowgraph (SFG)? SFG Topological network representation consisting of nodes & branches used to convert one form of network to a more suitable form (e.g. passive LC filters to integrator based filters) Any network described by a set of linear diff. equations can be expressed in SFG form. For a given network, many different SFGs exists. Choice of a particular SFG is based on practical considerations such as type of available components. ef: W.Heinlein & W. Holmes, Active Filters for Integrated Circuits, Prentice Hall, Chap. 8, 974. EECS 47 Lecture 3: Filters 4 H.K. Page 8

5 What is a Signal Flowgraph (SFG)? SFG nodes represent variables (V & I in our case), branches represent transfer functions (we will call these transfer functions branch multiplication factor BMF) To convert a network to its SFG form, KCL & KVL is used to derive state space description: Example: Circuit Statespace SFG Iin V o description Vo= Iin I in Io L Iin= Vo SL SL I o I in C = Io SC I in SC EECS 47 Lecture 3: Filters 4 H.K. Page 9 Signal Flowgraph (SFG) ules Two parallel branches can be replaced by a single branch with BMF equal to sum of two BMFs V b a V ab A node with only one incoming branch & one outgoing branch can be replaced by a single branch with BMF equal to the product of the two BMFs V a b V V a.b V V 3 An intermediate node can be multiplied by a factor (x). BMFs for incoming branches have to be multiplied by x and outgoing branches divided by x V a b V V x.a b/x V V 3 x.v 3 V V EECS 47 Lecture 3: Filters 4 H.K. Page

6 Signal Flowgraph (SFG) ules Simplifications can often be achieved by shifting or eliminating nodes V i V V 4 a /b V 3 V i V /b /b a V 3 A selfloop branch with BMF y can be eliminated by multiplying the BMF of incoming branches by /(y) /b /b V i V /b a V 3 V i V a/(/b) V 3 EECS 47 Lecture 3: Filters 4 H.K. Page Integrator Based Filters First Order LPF Start from C prototype Use KCL & KVL to derive state space description: V= Vo I Vo = sc V I = s I = I Use state space description to draw signal flowgraph (SFG) V s s I I SFG V C sc I I EECS 47 Lecture 3: Filters 4 H.K. Page

7 Normalize Since integrators the main building blocks require in & out signals in the voltage form (not current) Convert all currents to voltages by multiplying current nodes by a scaling resistance Branches should then be scaled accordingly V= Vo V I = s I V o = sc I = I I x = Vx V= Vo I = V s I Vo = sc I = I V= Vo V = V s V Vo = sc V = V EECS 47 Lecture 3: Filters 4 H.K. Page 3 Normalize V s sc V s sc V s sc I I I I V V EECS 47 Lecture 3: Filters 4 H.K. Page 4

8 Synthesis V s V V sc V V τ s V V τ = s, = C V τ s V EECS 47 Lecture 3: Filters 4 H.K. Page 5 First Order Integrator Based Filter V V τ s V H ( s) = τ s EECS 47 Lecture 3: Filters 4 H.K. Page 6

9 OpampC SingleEnded Integrator C V in Vo Vo= dt, = C sc τ = C EECS 47 Lecture 3: Filters 4 H.K. Page 7 OpampC Integrator Singleended OpampC integrator has a sign inversion from input to output Convert SFG accordingly by modifying BMF = EECS 47 Lecture 3: Filters 4 H.K. Page 8

10 OpampC st Order Filter C = Vo Vo = sc EECS 47 Lecture 3: Filters 4 H.K. Page 9 OpampC First Order Filter Noise Identify noise sources (here it is resistors & opamp) Find transfer function from each noise source to the output (opamp noise next page) n vo = m= H(f) m S(f)df i S(f) i Inputreferrednoisespectraldensity H(f) = H(f) = ( π fc) v n = v n = 4KT f v n v n C v o = kt C α = α Typically, α increases as filter order increases EECS 47 Lecture 3: Filters 4 H.K. Page

11 OpampC Filter Noise Opamp Contribution So far only the fundamental noise sources are considered. In reality, noise associated with the opamp increases the overall noise. The bandwidth of the opamp affects the opamp noise contribution to the total noise v n C v n v opamp Vo EECS 47 Lecture 3: Filters 4 H.K. Page State space description: V= VL = Vo IC VC = sc V I = VL IL = sl IC= Iin I IL Integrator Based Filters Second Order LC Filter Integrator form I in V I SFG V V C V L L V C C I L I C sc V L sl Draw signal flowgraph (SFG) I I in I C I L EECS 47 Lecture 3: Filters 4 H.K. Page

12 Normalize Convert currents to voltages by multiplying all current nodes by the scaling resistance V V C I I in sc I C V L sl I L I x = Vx V V V sc V V 3 sl EECS 47 Lecture 3: Filters 4 H.K. Page 3 Synthesis V V V sc V 3 sl τ sτ sτ τ = C = L EECS 47 Lecture 3: Filters 4 H.K. Page 4

13 Second Order Integrator Based Filter Filter Magnitude esponse V BP Magnitude (db) 5 5 sτ sτ V HP VLP. Normalized Frequency [Hz] EECS 47 Lecture 3: Filters 4 H.K. Page 5 Second Order Integrator Based Filter τ s βτ V V BP = in ττ s s VLP V = in ττ s βτ s VHP ττ s V = in ττ s βτ s τ= C τ= L β = ω = ττ = Q= β τ τ LC Frommatchingpointofviewdesirable: τ= τ Q = V BP sτ sτ V HP VLP EECS 47 Lecture 3: Filters 4 H.K. Page 6

14 Second Order Bandpass Filter Noise n vo = m= Find transfer function of each noise source to the output Integrate contribution of all noise sources Here it is assumed that opamps are noise free (not usually the case!) v n = v n = 4KT H m(f) S(f) i V BP v n sτ sτ v n v kt o = Q C α Typically, α increases as filter order increases Note the noise power is directly proportion to Q EECS 47 Lecture 3: Filters 4 H.K. Page 7 Second Order Integrator Based Filter Biquad By combining outputs can generate biquad function: V aττ s aτs a = 3 ττ s βτs a a a3 V BP jω splane sτ sτ σ V HP V LP EECS 47 Lecture 3: Filters 4 H.K. Page 8

15 Summary Integrator Based Monolithic Filters Signal flowgraph techniques utilized to convert LC networks to all integrator active filters Each reactive element (L& C) replaced by an integrator Fundamental noise limitation determined by integrating capacitor: For lowpass filter: Bandpass filter: v o = v o = kt α C α Q kt C where α is a function of filter order and topology EECS 47 Lecture 3: Filters 4 H.K. Page 9 Higher Order Filters Cascade of Biquads Example: LPF filter for CDMA X baseband LPF with fpass = 65 khz pass =. db fstop = 75 khz stop = 45 db 7th order Elliptic Filter Implementation with Biquads Goal: Maximize dynamic range Pair poles and zeros highest Q poles with closest zeros is a good starting point, but not necessarily optimum Ordering: Lowest Q poles first is a good start EECS 47 Lecture 3: Filters 4 H.K. Page 3

16 Filter Frequency esponse Phase (deg) Magnitude (db) kHz Bode Diagram MHz Frequency [Hz] 3MHz Mag. (db). EECS 47 Lecture 3: Filters 4 H.K. Page 3 PoleZero Map Imag Axis X splane PoleZero Map.5.5 eal Axis x 7 Q pole f pole [khz] f zero [khz] EECS 47 Lecture 3: Filters 4 H.K. Page 3

17 Biquad esponse.5 LPF Biquad Biquad Biquad EECS 47 Lecture 3: Filters 4 H.K. Page 33 Biquad esponse Bode Magnitude Diagram Magnitude (db) 3 LPF Biquad 4 Biquad 3 Biquad Frequency [Hz] EECS 47 Lecture 3: Filters 4 H.K. Page 34

18 Magnitude (db) Magnitude (db) khz Intermediate Outputs LPF 8 khz MHz 6 MHz khz khz MHz Frequency [Hz] Frequency [Hz] Magnitude (db) LPF Biquad LPF Biquads,3 LPF Biquads,3,4 Biquads,, 3, & Magnitude (db) MHz EECS 47 Lecture 3: Filters 4 H.K. Page 35 Sensitivity Component variation in Biquad 4 (highest Q pole): Increase w p4 by % Decrease w z4 by %.db Magnitude (db) 3 3dB 4 5 khz 6kHz Frequency [Hz] MHz High Q poles High sensitivity in Biquad realizations EECS 47 Lecture 3: Filters 4 H.K. Page 36

19 High Q & High Order Filters Cascade of biquads Highly sensitive to component variations not suitable for implementation of high Q & high order filters Cascade of biquads only used in cases where required Q for all biquads <4 (e.g. filters for disk drives) LC ladder filters more appropriate for high Q & high order filters (next topic) Less sensitive to component variations EECS 47 Lecture 3: Filters 4 H.K. Page 37 LC Ladder Filters s C L C3 L4 C5 L Made of resistors, inductors, and capacitors Doubly terminated or singly terminated Doubly terminated LC ladder filters Lowest sensitivity to component variations EECS 47 Lecture 3: Filters 4 H.K. Page 38

20 LC Ladder Filters s C L C3 L4 C5 L Design: CAD tools Matlab Spice Filter tables A. Zverev, Handbook of filter synthesis, Wiley, 967. A. B. Williams and F. J. Taylor, Electronic filter design, 3 rd edition, McGrawHill, 995. EECS 47 Lecture 3: Filters 4 H.K. Page 39 LC Ladder Filter Design Example Design a LPF with maximally flat passband: f3db = MHz, fstop = MHz s >7dB Maximally flat passband Butterworth fstop / f3db = s >7dB Determine minimum filter order : Use of Matlab or Tables 5th order Butterworth Stopband Attenuation db Νοrmalized w From: Williams and Taylor, p. 37 EECS 47 Lecture 3: Filters 4 H.K. Page 4

21 LC Ladder Filter Design Example Find values for L & C from Table: Note L &C values normalized to w 3dB = Denormalization: Multiply all L Norm, C Norm by: L r = /w 3dB C r = /(Xw 3dB ) is the value of the source and termination resistor (choose both Ω for now) Then: L= L r xl Norm C= C r xc Norm From: Williams and Taylor, p..3 EECS 47 Lecture 3: Filters 4 H.K. Page 4 LC Ladder Filter Design Example Find values for L & C from Table: Normalized values: C Norm =C5 Norm =.68 C3 Norm =. L Norm = L4 Norm =.68 Denormalization: Since w 3dB =πxmhz L r = /w 3dB = 4. nh C r = /(Xw 3dB )= 4. nf = C=C5=9.836nF, C3=3.83nF L=L4=5.75nH From: Williams and Taylor, p..3 EECS 47 Lecture 3: Filters 4 H.K. Page 4

22 Magnitude esponse Simulation L=5.75nH L4=5.75nH s=ohm C 9.836nF C3 3.83nF C nF L=Ohm 5 6 db passband attenuation due to double termination Magnitude (db) 3 4 3dB 5 3 Frequency [MHz] EECS 47 Lecture 3: Filters 4 H.K. Page 43 LC Ladder Filter Conversion to Integrator Based Active Filter V V V3 V4 V5 V6 s I L I3 L4 I 5 C C3 C5 I I4 I 6 I 7 L Use KCL & KVL to derive equations: I V = V, V =, V3 = V V sc 4 I I V =, V5 = V4 V 6, V6 = Vo = V sc sc V V3 I =, I = I I 3, I s 3 = sl V5 V I I I, I 6 4 = =, I6 = I5 I 7, I7 = sl4 L EECS 47 Lecture 3: Filters 4 H.K. Page 44

23 V I V s sc I LC Ladder Filter Signal Flowgraph I V = V in V, V =, V3 = V V sc 4 I4 I V, V V V, V 6 4 = 5 = = Vo = V sc sc V V3 I =, I = I I 3, I3 = s sl V5 V I I I, I 6 4 = =, I6 = I5 I 7, I7 = sl4 L V3 V4 V5 V6 sl sc3 sl4 sc5 I3 I4 I 5 I 6 I 7 SFG EECS 47 Lecture 3: Filters 4 H.K. Page 45 Vo L LC Ladder Filter Signal Flowgraph V V V3 V V 4 5 V6 s I L I3 L4 I 5 C C3 C5 I I4 I 6 I 7 L V I V s sc I V3 V4 V5 V6 sl sc3 sl4 sc5 Vo I3 I4 I 5 I 6 I 7 SFG L EECS 47 Lecture 3: Filters 4 H.K. Page 46

24 V I V s sc I LC Ladder Filter Normalize V3 V4 V5 V6 sl sc3 sl4 sc5 Vo I 3 I4 I 5 I 6 I 7 L V V s V V3 V4 V5 V6 sc sc sl 3 sl sc 4 5 V V 3 V 4 V 5 V 6 Vo V 7 L EECS 47 Lecture 3: Filters 4 H.K. Page 47 LC Ladder Filter Synthesize V V s V V3 V4 V5 V6 sc sc sl 3 sl sc 4 5 V V 3 V 4 V 5 V 6 Vo V 7 L s sτ sτ sτ 3 sτ 4 sτ 5 L EECS 47 Lecture 3: Filters 4 H.K. Page 48

25 s LC Ladder Filter Integrator Based Implementation sτ sτ sτ sτ 3 4 sτ 5 L L L4 τ = C., τ = = C., C., C., C. τ3= 3 τ4 = = 4 τ5 = 5 Building Block: C Integrator V V = sc EECS 47 Lecture 3: Filters 4 H.K. Page 49 Negative esistors EECS 47 Lecture 3: Filters 4 H.K. Page 5

26 Synthesize EECS 47 Lecture 3: Filters 4 H.K. Page 5 Frequency esponse EECS 47 Lecture 3: Filters 4 H.K. Page 5

27 Scale Node Voltages Scale by factor s EECS 47 Lecture 3: Filters 4 H.K. Page 53 Noise Total noise:.4 µv rms (noiseless opamps) That s excellent, but the capacitors are very large (and the resistors small). Not possible to integrate. Suppose our application calls for 4 µv rms EECS 47 Lecture 3: Filters 4 H.K. Page 54

28 Scale to Meet Noise Target Scale capacitors and resistors to meet noise objective s = 4 Noise: 4 µv rms (noiseless opamps) EECS 47 Lecture 3: Filters 4 H.K. Page 55 Completed Design EECS 47 Lecture 3: Filters 4 H.K. Page 56

29 Sensitivity C made (arbitrarily) 5% (!) larger than its nominal value.5 db error at band edge 3.5 db error in stopband Looks like very low sensitivity EECS 47 Lecture 3: Filters 4 H.K. Page 57 Differential Lowpass Filter Since each signal and its inverse readily available, eliminates the need for negative resistors! Differential design has the advantage of even order distortion and common mode spurious pickup automatically cancels Disadvantage: Double resistor and capacitor area! Overall ~3dB higher dynamic range compared to singleended EECS 47 Lecture 3: Filters 4 H.K. Page 58