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# L3 f10 2

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### L3 f10 2

1. 1. EECS 247 Lecture 3: Filters© 2010 H.K. Page 1EE247Administrative –Homework #1 will be posted on EE247 site and is due Sept. 9th –Office hours held @ 201 Cory Hall: •Tues. and Thurs.: 4 to 5pmEECS 247 Lecture 3: Filters © 2010 H.K. Page 2 EE247 Lecture 3 •Active Filters –Active biquads- –How to build higher order filters? •Integrator-based filters –Signal flowgraph concept –First order integrator-based filter –Second order integrator-based filter & biquads –High order & high Q filters •Cascaded biquads & first order filters –Cascaded biquad sensitivityto component mismatch •Ladder type filters
2. 2. EECS 247 Lecture 3: Filters © 2010 Page 3 Filters 2nd Order Transfer Functions (Biquads) • Biquadratic (2nd order) transfer function: P H j Q H j H j P              ( ) ( ) 0 ( ) 1 0 2 2 P P P 2 2 2 2 P P P P 2 P P 1 P 2 1 H( s ) s s 1 Q 1 H( j ) 1 Q Biquad poles @: s 1 1 4Q 2Q Note: for Q poles are real , comple x otherwise                                       EECS 247 Lecture 3: Filters © 2010 Page 4 Biquad Complex Poles Distance from origin in s-plane:   2 2 2 2 1 4 1 2 P P P P Q Q d               1 2 2 Complex conjugate poles: s 1 4 1 2 P P P P Q j Q Q              poles j  d S-plane
3. 3. EECS 247 Lecture 3: Filters © 2010 Page 5 s-Plane poles j  P radius   P 2Q real part - P   P 2Q 1 arccos 2 s 1 4 1 2 P P P j Q Q            EECS 247 Lecture 3: Filters © 2010 Page 6 Example 2nd Order Butterworth j  45 deg 2 2 4 2 2 Since 2nd order Butterworth: ree Cos p Q        P 2Q 1 arccos
5. 5. EECS 247 Lecture 3: Filters © 2010 Page 9 Addition of Imaginary Axis Zeros • Sharpen transition band • Can “notch out” interference – Band-reject filter • High-pass filter (HPF) Note: Always represent transfer functions as a product of a gain term, poles, and zeros (pairs if complex). Then all coefficients have a physical meaning, and readily identifiable units. 2 Z 2 P P P 2 P Z s 1 H( s ) K s s 1 Q H( j ) K                              EECS 247 Lecture 3: Filters © 2010 Page 10 Imaginary Zeros • Zeros substantially sharpen transition band • At the expense of reduced stop-band attenuation at high frequencies Z P P P f f Q f kHz 3 2 100    10 4 10 5 10 6 10 7 -50 -40 -30 -20 -10 0 10 Frequency [Hz] Magnitude [dB] With zeros No zeros Real Axis Imag Axis -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x 10 6 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x 10 6 Pole-Zero Map
6. 6. EECS 247 Lecture 3: Filters © 2010 Page 11 Moving the Zeros Z P P P f f Q f kHz    2 100 104 105 106 107 -50 -40 -30 -20 -10 0 10 20 Frequency [Hz] Magnitude [dB] Pole-Zero Map Real Axis Imag Axis -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 x10 5 x10 5 EECS 247 Lecture 3: Filters © 2010 Page 12 Tow-Thomas Active Biquad Ref: P. E. Fleischer and J. Tow, “Design Formulas for biquad active filters using three operational amplifiers,” Proc. IEEE, vol. 61, pp. 662-3, May 1973. • Parasitic insensitive • Multiple outputs
7. 7. EECS 247 Lecture 3: Filters © 2010 Page 13 Frequency Response         1 0 2 0 2 0 1 0 0 1 1 0 3 1 0 2 1 0 2 2 2 1 0 2 2 1 1 2 0 0 2 1 1 s a s a b b a s a b a b V k a V s a s a b s b s b V V s a s a b a b s b a b k V V in o in o in o                    • Vo2 implements a general biquad section with arbitrary poles and zeros • Vo1 and Vo3 realize the same poles but are limited to at most one finite zero • Possible to use combination of 3 outputs EECS 247 Lecture 3: Filters © 2010 Page 14 Component Values 8 7 2 3 7 1 2 8 2 1 1 1 1 2 3 7 1 2 8 0 6 8 2 4 7 1 8 6 8 1 1 1 3 5 7 1 2 8 0 1 1 R R k R R C R R C k R C a R R R C C R a R R b R R R R R R R C b R R R C C R b                 7 2 8 2 8 6 0 2 1 0 5 2 1 2 1 1 4 1 2 0 1 3 0 2 1 2 1 1 1 1 1 1 1 1 1 R k R b R R b C k a R k a b b C R k k a C R a C k R a C R         1 2 8 given a ,b ,k , C , C and R i i i it follows that 1 1 2 3 7 1 2 8 Q RC R R R C C R P P P    
8. 8. EECS 247 Lecture 3: Filters© 2010 H.K. Page 15Higher-Order Filters in the Integrated Form •One way of building higher-order filters (n>2) is via cascade of 2ndorder biquads & 1storder , e.g. Sallen-Key,or Tow-Thomas, & RC2ndorderFilter …… Nx 2ndorder sections Filter order: n=2N 1 2 NCascade of 1stand 2ndorder filters:  Easy to implement  Highly sensitive to component mismatch -good for low Q filters only For high Q applications good alternative: Integrator-based ladder filters2ndorderFilter 1stor 2ndorderFilter EECS 247 Lecture 3: Filters© 2010 H.K. Page 16 Integrator Based Filters •Main building block for this category of filters Integrator •By using signal flowgraphtechniques Conventional RLC filter topologies can be converted to integrator based type filters •How to design integrator based filters? –Introduction to signal flowgraphtechniques –1st order integrator based filter –2nd order integrator based filter –High order and high Q filters
9. 9. EECS 247 Lecture 3: Filters © 2010 H.K. Page 17 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 RLC 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 *Ref: W.Heinlein & W. Holmes, “Active Filters for Integrated Circuits”, Prentice Hall, Chap. 8, 1974. EECS 247 Lecture 3: Filters © 2010 H.K. Page 18 What is a Signal Flowgraph (SFG)? • Signal flowgraph technique consist of nodes & branches: – Nodes represent variables (V & I in our case) – Branches represent transfer functions (we will call the transfer function branch multiplication factor or BMF) • To convert a network to its SFG form, KCL & KVL is used to derive state space description • Simple example: Circuit State-space description SFG Z Z Iin Vo Iin Vo IinZ Vo
10. 10. EECS 247 Lecture 3: Filters © 2010 H.K. Page 19 Signal Flowgraph (SFG) Examples 1 SL Circuit State-space description SFG R L R Vo C 1 SC Vin Vo Io Iin Vo Iin Iin Iin Vo Vin Io Iin R Vo 1 Vin Io SL 1 Iin Vo SC       EECS 247 Lecture 3: Filters © 2010 H.K. Page 20 Useful Signal Flowgraph (SFG) Rules V1 a V2 b a+b V1 V2 a.b V1 V2 a V3 b V1 2 V a.V1+b.V1=V2 (a+b).V1=V2 a.V1=V3 (1) b.V3=V2 (2) Substituting for V3 from (1) in (2) (a.b).V1=V2 • Two parallel branches can be replaced by a single branch with overall BMF equal to sum of two BMFs •A node with only one incoming branch & one outgoing branch can be eliminated & replaced by a single branch with BMF equal to the product of the two BMFs
11. 11. EECS 247 Lecture 3: Filters © 2010 H.K. Page 21 Useful Signal Flowgraph (SFG) Rules •An intermediate node can be multiplied by a factor (k). BMFs for incoming branches have to be multiplied by k and outgoing branches divided by k V3 a b V1 2 V k.a b/k V1 V2 k.V3 a.V1=V3 (1) b.V3=V2 (2) Multiply both sides of (1) by k (a.k) . V1= k.V3 (1) Divide & multiply left side of (2) by k (b/k) . k.V3 = V2 (2) EECS 247 Lecture 3: Filters © 2010 H.K. Page 22 Useful Signal Flowgraph (SFG) Rules h Vi V2 a Vo b -b g V3 h Vi V2 a/(1+b) Vo b g V3 V3 c Vi V2 a Vo b c -b d Vi V2 a Vo -1 -b 1 d V3 • Simplifications can often be achieved by shifting or eliminating nodes • Example: eliminating node V4 • A self-loop branch with BMF y can be eliminated by multiplying the BMF of incoming branches by 1/(1-y) V4
12. 12. EECS 247 Lecture 3: Filters © 2010 H.K. Page 23 Integrator Based Filters 1st Order LPF • Conversion of simple lowpass RC filter to integrator-based type by using signal flowgraph techniques in Vo 1 V 1 s RC   Vo Rs C Vin EECS 247 Lecture 3: Filters © 2010 H.K. Page 24 What is an Integrator? Example: Single-Ended Opamp-RC Integrator Vo C Vin - + R a   in osC , o in o in V 1 1 V V V , V V dt R sRC RC         • Node x: since opamp has high gain Vx=-Vo /a 0 • Node x is at “virtual ground”  No voltage swing at Vx combined with high opamp input impedance  No input opamp current Vx IR IC
13. 13. EECS 247 Lecture 3: Filters © 2010 H.K. Page 25 What is an Integrator? Example: Single-Ended Opamp-RC Integrator Vin -  Note: Practical integrator in CMOS technology has input & output both in the form of voltage and not current  Consideration for SFG derivation Vo   RC o in V 1 V sRC   0 1 RC   -90o 20log H  0dB Phase   Ideal Magnitude & phase response EECS 247 Lecture 3: Filters © 2010 H.K. Page 26 1st Order LPF Convert RC Prototype to Integrator Based Version 1. Start from circuit prototype- Name voltages & currents for all components 2. Use KCL & KVL to derive state space description in such a way to have BMFs in the integrator form:  Capacitor voltage expressed as function of its current VCap.=f(ICap.)  Inductor current as a function of its voltage IInd.=f(VInd.) 3. Use state space description to draw signal flowgraph (SFG) (see next page) I1 Vo Rs C Vin I2 V1  VC  
14. 14. EECS 247 Lecture 3: Filters © 2010 H.K. Page 27 Integrator Based Filters First Order LPF I1 Vo 1 Rs Vin C I2 V1  1 Rs 1 sC I1 I2 Vin 1 V1 1 VC • All voltages & currents  nodes of SFG • Voltage nodes on top, corresponding current nodes below each voltage node SFG VC   1 Vo V1 Vin VC 1 VC I2 sC Vo VC 1 I1 V1 Rs I2 I1         EECS 247 Lecture 3: Filters © 2010 H.K. Page 28 Normalize • Since integrators are the main building blocks require in & out signals in the form of voltage (not current)  Convert all currents to voltages by multiplying current nodes by a scaling resistance R*  Corresponding BMFs should then be scaled accordingly 1 in o 1 1 2 o 2 1 V V V V I Rs I V sC I I      1 in o * * 1 1 * 2 o * * * 2 1 V V V R I R V Rs I R V sC R I R I R      * ' IxR Vx 1 in o * ' 1 1 ' 2 o * ' ' 2 1 V V V R V V Rs V V sC R V V     
15. 15. EECS 247 Lecture 3: Filters © 2010 H.K. Page 29 1st Order Lowpass Filter SGF Normalize ' V2 * 1 sCR Vo Vin 1 V1 1 1 * R Rs 1 Rs 1 sC I1 I2 Vin 1 1 Vo 1 V1 ' V1 Vin 1 V1 1 Vo * R Rs I1 R   I2 R   * 1 sCR * * R R EECS 247 Lecture 3: Filters © 2010 H.K. Page 30 1st Order Lowpass Filter SGF Synthesis ' V1 1 * 1 sC R Vin 1 V1 1 Vo ' 1 V2 * R Rs * * R  Rs ,   R C ' V1 1  s Vin 1 V1 1 Vo ' 1 V2 1  s Vo Vin 1 V1 1 ' V2 1 Consolidate two branches * Choosing R  Rs
16. 16. EECS 247 Lecture 3: Filters © 2010 H.K. Page 31 First Order Integrator Based Filter Vo Vin - +   1 H s  s    + 1  s Vo Vin 1 V1 1 ' V2 1 EECS 247 Lecture 3: Filters © 2010 H.K. Page 32 1st Order Filter Built with Opamp-RC Integrator Vo Vin - +  + • Single-ended Opamp-RC integrator has a sign inversion from input to output  Convert SFG accordingly by modifying BMF Vo Vin -  + - Vo ' Vin  Vin +  + -
17. 17. EECS 247 Lecture 3: Filters © 2010 H.K. Page 33 1st Order Filter Built with Opamp-RC Integrator • To avoid requiring an additional opamp to perform summation at the input node: Vo ' Vin  Vin +  + - Vo ' Vin  - - EECS 247 Lecture 3: Filters © 2010 H.K. Page 34 1st Order Filter Built with Opamp-RC Integrator (continued) o ' in V 1 V 1 sRC    Vo C in ' V - + R R - - Vo ' Vin 
18. 18. EECS 247 Lecture 3: Filters © 2010 H.K. Page 35 Opamp-RC 1st Order Filter Noise 2 vn1   k 2 2 o 0 m m m 1 th i 2 2 1 2 2 2 2 n1 n2 2 o v H ( f ) S ( f ) df S ( f ) Noise spectral densi ty of i noise sou rce 1 H ( f ) H ( f ) 1 2 fRC v v 4KTR f k T v 2 C 2                 Vo C - + R R Typically,  increases as filter order increases Identify noise sources (here it is resistors & opamp) Find transfer function from each noise source to the output (opamp noise next page) 2 vn2  EECS 247 Lecture 3: Filters © 2010 H.K. Page 36 Opamp-RC Filter Noise Opamp Contribution 2 vn1 2 vopamp Vo C - + R R • So far only the fundamental noise sources are considered • In reality, noise associated with the opamp increases the overall noise • For a well-designed filter opamp is designed such that noise contribution of opamp is negligible compared to other noise sources • The bandwidth of the opamp affects the opamp noise contribution to the total noise 2 vn2
19. 19. EECS 247 Lecture 3: Filters © 2010 H.K. Page 37 Integrator Based Filter 2nd Order RLC Filter Vo 1 1 sL Iin R C •State space description: R L C o C C R R L L C in R L V V V V I V sC V I R V I sL I I I I          • Draw signal flowgraph (SFG) SFG L 1 R 1 sC IR IC VC Iin 1 1 VR 1 1 VL IL VC IL + - IC VL + - Integrator form + - VR IR EECS 247 Lecture 3: Filters © 2010 H.K. Page 38 1 1 * R R * 1 sCR ' V1 V2 Vin 1 1 V1 * R sL 1 Vo ' V3 2nd Order RLC Filter SGF Normalize 1 1 R 1 sC IR IC VC Iin 1 1 VR 1 1 VL IL 1 sL • Convert currents to voltages by multiplying all current nodes by the scaling resistance R* * ' IxR Vx ' V2
20. 20. EECS 247 Lecture 3: Filters © 2010 H.K. Page 39 Vin 1 * R R 2 1 s 1 1 s 1 1 * R R * 1 sCR ' V1 V2 Vin 1 1 V1 * R sL 1 Vo ' V3 2nd Order RLC Filter SGF Synthesis  Vo - - * * 1  R C  2  L R EECS 247 Lecture 3: Filters © 2010 H.K. Page 40 -20 -15 -10 -5 0 0.1 1 10 Second Order Integrator Based Filter Normalized Frequency [Hz] Magnitude (dB) Vin 1 * R R 2 1 s 1 1 s VBP - - VLP VHP  Filter Magnitude Response
21. 21. EECS 247 Lecture 3: Filters © 2010 H.K. Page 41 Second Order Integrator Based Filter Vin 1 * R R 2 1 s 1 1 s VBP - - VLP VHP  BP 2 in 2 1 2 2 LP in 2 1 2 2 2 HP 1 2 in 2 1 2 2 * * 1 2 * 0 1 2 1 2 1 2 * V s V s s 1 V 1 V s s 1 V s V s s 1 R C L R R R 1 1 Q 1 From matching pointof viewdes i rable : Q R R L C                                               EECS 247 Lecture 3: Filters © 2010 H.K. Page 42 Second Order Bandpass Filter Noise 2 2 n1 n2 2 o v v 4KTRdf k T v 2 Q C    Vin 1 * R R 2 1 s 1 1 s VBP - - 2 vn1 2 vn2 • 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!) k 2 2 o 0 m m m 1 v H ( f ) S ( f ) df      Typically,  increases as filter order increases  Note the noise power is directly proportion to Q 
22. 22. EECS 247 Lecture 3: Filters © 2010 H.K. Page 43 Second Order Integrator Based Filter Biquad Vin 1 * R R 2 1 s 1 1 s VBP - - 2 0 1 1 2 2 2 3 in 2 1 2 2 V a s a s a V s  s 1             Vo  a1 a2 a3 VHP VLP • By combining outputs can generate general biquad function: s-plane j  EECS 247 Lecture 3: Filters © 2010 H.K. Page 44 Summary Integrator Based Monolithic Filters • Signal flowgraph techniques utilized to convert RLC networks to integrator based active filters • Each reactive element (L& C) replaced by an integrator • Fundamental noise limitation determined by integrating capacitor value: – For lowpass filter: – Bandpass filter: where  is a function of filter order and topology 2 o k T v Q C   2 o k T v C  