# 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

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

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

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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