Integrator Based Filters
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1 Integrator Based Filters Main building block for this category of filters integrator By using signal flowgraph techniques conventional filter topologies can be converted to integrator based type filters Next few pages: Signal flowgraph techniques st order integrator based filter nd order integrator based filter High order and high Q filters EECS 47 Lecture 3: Filters 5 H.K. Page 5 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 differential 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 5 H.K. Page 6
2 What is a Signal Flowgraph (SFG)? SFG nodes represent variables ( & 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 & KL is used to derive state space description: Example: Circuit Statespace SFG Iin description I in o Iin o I o L Io SL SL I o I in C Iin o SC I in SC EECS 47 Lecture 3: Filters 5 H.K. Page 7 Signal Flowgraph (SFG) ules Two parallel branches can be replaced by a single branch with overall BMF equal to sum of two BMFs b a 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 a b a.b 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 a b x.a b/x 3 x. 3 EECS 47 Lecture 3: Filters 5 H.K. Page 8
3 Signal Flowgraph (SFG) ules Simplifications can often be achieved by shifting or eliminating nodes i 4 a /b 3 i /b /b a 3 A selfloop branch with BMF y can be eliminated by multiplying the BMF of incoming branches by /(y) /b /b i /b a 3 i a/(/b) 3 EECS 47 Lecture 3: Filters 5 H.K. Page 9 Integrator Based Filters st Order LPF Start from C prototype Use KCL & KL to derive state space description: s I C I o C Use state space description to draw signal flowgraph (SFG) EECS 47 Lecture 3: Filters 5 H.K. Page
4 Integrator Based Filters First Order LPF KCL & KL to derive state space description: I C C I s I I o C Use state space description to draw signal flowgraph (SFG) I s s I C I SFG C C I EECS 47 Lecture 3: Filters 5 H.K. Page 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 Corresponding BMFs should then be scaled accordingly o I s I o I I Ix x o I s I o I I o s o EECS 47 Lecture 3: Filters 5 H.K. Page
5 Normalize s s s I I I I EECS 47 Lecture 3: Filters 5 H.K. Page 3 Synthesis s τ s Consolidate two branches τ s, C τ s EECS 47 Lecture 3: Filters 5 H.K. Page 4
6 First Order Integrator Based Filter τ s H ( s) τ s EECS 47 Lecture 3: Filters 5 H.K. Page 5 OpampC SingleEnded Integrator C in o o dt, C τ C EECS 47 Lecture 3: Filters 5 H.K. Page 6
7 State space description: L C o IC C I IL L sl IC Iin I IL Integrator Based Filter nd Order LC Filter Integrator form I in I SFG C L L C I L C I C L sl Draw signal flowgraph (SFG) I I in I C I L EECS 47 Lecture 3: Filters 5 H.K. Page 3 Normalize Convert currents to voltages by multiplying all current nodes by the scaling resistance C L sl I x x sl I I in I C I L 3 EECS 47 Lecture 3: Filters 5 H.K. Page 3
8 Synthesis 3 sl τ sτ sτ τ C L EECS 47 Lecture 3: Filters 5 H.K. Page 33 Second Order Integrator Based Filter Filter Magnitude esponse BP Magnitude (db) 5 5 sτ sτ HP LP. Normalized Frequency [Hz] EECS 47 Lecture 3: Filters 5 H.K. Page 34
9 BP τs in ττ βτ s LP in ττ βτ s HP ττ s in ττ βτ s τ C τ L β ω ττ LC Q β Second Order Integrator Based Filter BP sτ sτ HP LP τ τ Frommatchingpointofviewdesirable: τ τ Q EECS 47 Lecture 3: Filters 5 H.K. Page 35 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!) vn vn 4KTdf H m(f) S(f)df i BP v n sτ sτ v n vo kt Q C α Typically, α increases as filter order increases Note the noise power is directly proportion to Q EECS 47 Lecture 3: Filters 5 H.K. Page 36
10 Second Order Integrator Based Filter Biquad By combining outputs can generate general biquad function: aττ s aτs a 3 ττ s βτs a a a3 BP jω splane sτ sτ σ HP LP EECS 47 Lecture 3: Filters 5 H.K. Page 37 Summary Integrator Based Monolithic Filters Signal flowgraph techniques utilized to convert LC networks to integrator based active filters Each reactive element (L& C) replaced by an integrator Fundamental noise limitation determined by integrating capacitor: For lowpass filter: Bandpass filter: vo vo kt α C kt α Q C where α is a function of filter order and topology EECS 47 Lecture 3: Filters 5 H.K. Page 38
11 Higher Order Filters How do we build higher order filters? Cascade of biquads and st order sections Each complex conjugate pole built with a biquad and real pole with st order section Easy to implement In the case of high order high Q filters highly sensitive to component variations Direct conversion of high order ladder type LC filters SFG techniques used to perform exact conversion of ladder type filters to integrator based filters More complicated conversion process Much less sensitive to component variations compared to cascade of biquads EECS 47 Lecture 3: Filters 5 H.K. Page 39 Higher Order Filters Cascade of Biquads Example: LPF filter for CDMA baseband receiver LPF with fpass 65 khz pass. db fstop 75 khz stop 45 db Assumption: Can compensate for phase distortion in the digital domain 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 5 H.K. Page 4
12 Filter Frequency esponse Bode Diagram Phase (deg) Magnitude (db) kHz MHz Frequency [Hz] 3MHz Mag. (db). EECS 47 Lecture 3: Filters 5 H.K. Page 4 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 5 H.K. Page 4
![Page 4 PoleZero Map Imag Axis X 7.5.5 splane PoleZero Map.5.5 eal Axis x 7 Q pole f pole [khz] 6.](/docs-images/41/22510068/images/page_12.jpg "79 659.496 3.659 6.744.6 473.643 39.568 f zero [khz] 97.5 836.6 744.")
13 Biquad esponse.5 LPF Biquad Biquad Biquad EECS 47 Lecture 3: Filters 5 H.K. Page 43 Biquad esponse Bode Magnitude Diagram Magnitude (db) 3 4 LPF Biquad Biquad 3 Biquad Frequency [Hz] EECS 47 Lecture 3: Filters 5 H.K. Page 44
14 Magnitude (db) Magnitude (db) khz Intermediate Outputs LPF Magnitude (db) Magnitude (db) LPF Biquad LPF Biquads,3 LPF Biquads,3,4 Biquads,, 3, & khz MHz 6 MHz khz khz MHz MHz Frequency [Hz] Frequency [Hz] EECS 47 Lecture 3: Filters 5 H.K. Page 45 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 5 H.K. Page 46
15 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 5 H.K. Page 47 Ladder Type Filters For simplicity, will start with all pole ladder type filters Convert to integrator based form Example shown Then will attend to high order ladder type filters incorporating zeros Implement the same 7 th order elliptic filter in the form of ladder type Find level of sensitivity to component variations Compare with cascade of biquads Convert to integrator based form utilizing SFG techniques Example shown EECS 47 Lecture 3: Filters 5 H.K. Page 48
16 LC Ladder Filters s C L C3 L4 C5 L Made of resistors, inductors, and capacitors Doubly terminated or singly terminated (with or w/o L ) Doubly terminated LC ladder filters Lowest sensitivity to component variations EECS 47 Lecture 3: Filters 5 H.K. Page 49 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 5 H.K. Page 5
17 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 5.9 nh C r /(Xw 3dB ) 5.9 nf CC59.836nF, C33.83nF LL45.75nH From: Williams and Taylor, p..3 EECS 47 Lecture 3: Filters 5 H.K. Page 53 Magnitude esponse Simulation sohm L5.75nH C 9.836nF L45.75nH C3 3.83nF C nF LOhm 5 SPICE simulation esults 6 db passband attenuation due to double termination Magnitude (db) 3 4 3dB 5 3 Frequency [MHz] EECS 47 Lecture 3: Filters 5 H.K. Page 54
18 LC Ladder Filter Conversion to Integrator Based Active Filter s I L I3 L4 I 5 C C3 C5 I I4 I 6 I 7 L Use KCL & KL to derive equations: I,, 3 4 I I 4 6 4, 5 4 6, 6 o I, I I I 3, I s 3 sl 5 6 I 4 I 3 I 5, I 5, I 6 I 5 I 7, I7 sl4 L EECS 47 Lecture 3: Filters 5 H.K. Page 55 I s I LC Ladder Filter Signal Flowgraph I in,, 3 4 I 4 I,, o I, I I I 3, I3 s sl 5 6 I 4 I 3 I 5, I 5, I 6 I 5 I 7, I7 sl4 L sl 3 sl4 5 I3 I4 I 5 I 6 I 7 SFG EECS 47 Lecture 3: Filters 5 H.K. Page 56 o L
19 LC Ladder Filter Signal Flowgraph s I L I3 L4 I 5 C C3 C5 I I4 I 6 I 7 L I s I sl 3 sl4 5 o I3 I4 I 5 I 6 I 7 SFG L EECS 47 Lecture 3: Filters 5 H.K. Page 57 I s I LC Ladder Filter Normalize sl 3 sl4 5 o I 3 I4 I 5 I 6 I 7 L s sl 3 sl o 7 L EECS 47 Lecture 3: Filters 5 H.K. Page 58
20 s LC Ladder Filter Synthesize sl 3 sl o 7 L s sτ sτ sτ 3 sτ 4 sτ 5 L EECS 47 Lecture 3: Filters 5 H.K. Page 59 s LC Ladder Filter Integrator Based Implementation sτ sτ sτ 3 sτ 4 sτ 5 L L L4 C., C., C., C., C τ τ τ τ τ Building Block: C Integrator EECS 47 Lecture 3: Filters 5 H.K. Page 6
21 Negative esistors o o o EECS 47 Lecture 3: Filters 5 H.K. Page 6 Synthesize EECS 47 Lecture 3: Filters 5 H.K. Page 6
22 Frequency esponse EECS 47 Lecture 3: Filters 5 H.K. Page 63 Scale Node oltages Scale by factor s EECS 47 Lecture 3: Filters 5 H.K. Page 64
23 Noise Total noise:.4 µ rms (noiseless opamps) That s excellent, but the capacitors are very large (and the resistors small). Not possible to integrate. Suppose our application allows higher noise in the order of 4 µ rms EECS 47 Lecture 3: Filters 5 H.K. Page 65 Scale to Meet Noise Target Scale capacitors and resistors to meet noise objective s 4 Noise: 4 µ rms (noiseless opamps) EECS 47 Lecture 3: Filters 5 H.K. Page 66
24 Completed Design EECS 47 Lecture 3: Filters 5 H.K. Page 67 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 5 H.K. Page 68
25 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 4: Filters 5 H.K. Page 3 Differential 5 th Order Lowpass Filter Since each signal and its inverse readily available, eliminates the need for negative resistors! Differential design has the advantage of even order harmonic distortion and common mode spurious pickup automatically cancels Disadvantage: Double resistor and capacitor area! EECS 47 Lecture 4: Filters 5 H.K. Page 4
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