HO4_315aSP09_biquads

HO4_315aSP09_biquads - Biquad Realization Boris Murmann...

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Unformatted text preview: Biquad Realization Boris Murmann Stanford University murmann@stanford.edu Copyright © 2009 by Boris Murmann B. Murmann EE315A ― HO #4 1 Architectural Options for High-Order Filters • Cascades of (active) first and second-order sections (biquads) • Ladder filters (passive or emulated using active components) • Specialized architectures, typically emphasizing low complexity – Watch out for sensitivity issues B. Murmann EE315A ― HO #4 2 Building Block Options [Kuhn, IEEE TCAS II, 10/2003] B. Murmann EE315A ― HO #4 3 Example • An interesting filter that combines three different approaches – Passive LC – Active RC – Switched capacitor [Schreier, JSSC 12/2002] B. Murmann EE315A ― HO #4 4 The Challenge • Way too many options available y y p • Deciding on which implementation is “best” may only be possible once several options have been thoroughly compared – I terms of both first-order properties and second-order In t f b th fi t d ti d d d nonidealities, which aren’t always easy to understand • The following discussion starts from the most basic ideas, and derives some of the most popular solutions used in practice • For now, we will focus on the realization of biquads, and cover ladder based ladder-based implementations next – The treatment of biquads will help us understand why we may ultimately want go for a ladder implementation EE315A ― HO #4 B. Murmann 5 Passive LC Lowpass Filter H( s ) = 1 sC 1 + R + sL sC ωP = B. Murmann = 1 2 1 + sRC + s LC 1 LC QP = R EE315A ― HO #4 1 = 1+ s s2 + 2 ωPQP ωP L C 6 On-chip Capacitors Metal-Insulator-Metal (MIM) Vertical Parallel Plate (VPP) [Ng, Trans. Electron Dev. 7/2005] [Aparicio, JSSC 3/2002] • Typically 1-2 fF/μm2 (10-20 fF/μm2 for advanced structures) – For 1 fF/μm2, a 10 pF capacitor occupies ~100μm x 100μm • Both MIM and VPP capacitors have good electrical properties – Mostly worry about parasitic caps – Series and parallel resistances are often not a concern B. Murmann EE315A ― HO #4 7 On-chip Inductors ~1mm [Mohan, JSSC 10/1999] [Bevilacqua, ISSCC 2004] • Many nonidealities, hard to model, low “Q” y , , Q • Area inefficient, typically achieve L < 10nH • Sometimes bondwires are used as an alternative, L ~ 1nH/mm B. Murmann EE315A ― HO #4 8 Inductor Quality Factor Y= In general ⇒ Q= 1 Rs + j ωL ⇒ QL = Y= • X( ω) R 1 R + jX( ω) ωL Rs On-chip inductors typically achieve QL < 5-10 at their usable frequencies EE315A ― HO #4 B. Murmann 9 LC Lowpass Example • Assuming that we (very generously) use C=100pF, L=10nH ωP = 1 = 2π ⋅ 160MHz LC • Integrated p g passive LC filters become p practical for f > 200-500MHz • For our LC lowpass, if we assume R=Rs (i.e. we only use the parasitic resistor of the inductor no explicitly added resistance) inductor, QL = • ωL Rs QP = ω 1 L QL L = = QL P R C ωL C ω This means that at ω=ωP, the magnitude peaking that we can get is limited to the QL of the inductor (~5-10); not all that great B. Murmann EE315A ― HO #4 10 Summary • On-chip capacitors are g p p great, even though they’re usually not as , g y y large as we would like them to be • On-chip inductors tend to be avoided whenever possible, and are typically not useful in a filter with poles at frequencies below < 200-500 MHz • The solution to this problem is to “simulate” the inductors using active components ti t EE315A ― HO #4 B. Murmann 11 Gyrators • Gyrators are “active inductors” T. L. Deliyannis, J. K. Fidler and Y. Sun, Continuous-Time Active Filter Design g http://www.engnetbase.com/ej ournals/books/book_summary/ summary.asp?id=475 (Section 3.5) • The above circuit is not all that useful for our lowpass filter; we f f f need a “floating” inductor − Don’t want the inductance to be ground referenced B. Murmann EE315A ― HO #4 12 Floating Gyrator http://www.engnetbase.com/ej ournals/books/book_summary/ summary.asp?id=475 (Section 6.4) • Floating gyrators are pretty complex – There must be a better way to solve this problem problem… B. Murmann EE315A ― HO #4 13 Integrators • A circuit that we can build without much sweat is an active integrator, e.g. using an op-amp – Many more options exist (more later) • Assuming the availability of an ideal op-amp, we have vout (t ) = − 1 v in (t ) dt C∫ R Vout ( s ) = − B. Murmann EE315A ― HO #4 1 Vin ( s ) sRC 14 State-Space Realization State variables (integrator outputs) L vc (t ) = 1 ic (t )dt C∫ iL (t ) = 1 v L (t )dt L∫ Vc ( s ) = 1 Ic ( s ) sC IL ( s ) = 1 V (s) sL L L in C C out Vc = IL = B. Murmann 1 1 Ic = IL = Vout sC sC 1 1 VL = (Vin − ILR − Vout ) sL sL EE315A ― HO #4 15 Block Diagram • Looks promising, but the problem with this realization is that the first integrator takes a voltage at the input and produces a current at the output – We need the opposite if we want to realize the circuit with an RC integrator B. Murmann EE315A ― HO #4 16 Modified (Equivalent) Block Diagrams EE315A ― HO #4 B. Murmann 17 Implementation A1 A2 H( s ) = • A3 Vout ( s ) 1 =− Vout ( s ) 1 + sRC + s 2LC One remaining issue is that the transfer function is inverted – We could fix that (if needed) using a fourth op-amp – Or by pushing A2 toward the input, and utilizing both its inverting and non-inverting input • The latter trick is used in the so-called KHN biquad B. Murmann EE315A ― HO #4 18 KHN Biquad W.J. Kerwin, L.P. Huelsman, R.W Newcomb, "State-Variable Synthesis for Insensitive Integrated Circuit Transfer , , , y g Functions," IEEE JSSC, vol.2, no.3, pp. 87-92, Sep. 1967. http://www.engnetbase.com/ejournals/books/book_summary/summary.asp?id=475 (Section 4.9) EE315A ― HO #4 B. Murmann 19 Highpass and Bandpass Output • An interesting feature of some biquads (including the HKN) is that they provide additional highpass and bandpass outputs for “free” τ s 2 τ2 HHP ( s ) = 1+ B. Murmann s s2 + 2 ωPQP ωP τ −s τ HBP ( s ) = 1+ s s2 + 2 ωPQP ωP EE315A ― HO #4 1 HLP ( s ) = 1+ s s2 + 2 ωPQP ωP 20 The General HKN Biquad HP LP BP GENERAL HGENERAL ( s ) = B. Murmann b2s 2 + b1s + b0 s s2 + 2 1+ ωPQP ωP Implements arbitrary poles and zeros EE315A ― HO #4 21 Tow-Thomas Biquad P. E. Fleischer and J. Tow, “Design Formulas for biquad active filters using three operational amplifiers,” P lifi ” Proc. IEEE vol. 61 pp. 662 3 M 1973 IEEE, l 61, 662-3, May 1973. • General biquad using only three op-amps B. Murmann EE315A ― HO #4 22 Transfer Functions ( b a − b ) s + ( b2a0 − b0 ) Vo1 = −k 2 2 1 2 1 Vin s + a1s + a0 Vo 2 b2s 2 + b1s + b0 = 2 Vin s + a1s + a0 Vo 3 1 ( b0 − b2a0 ) s + ( a1b0 − a0b1 ) =− Vin s 2 + a1s + a0 k1 a0 • Vo2/Vin implements a general biquad section with arbitrary poles and zeros • Vo1/Vin and Vo3/Vin realize the same poles but are limited to at most one finite zero EE315A ― HO #4 B. Murmann 23 Design Equations b0 = b1 = b2 = a0 = a1 = k1 = k2 = B. Murmann R8 R3R5R7C1C2 1 ⎛ R8 R1R8 ⎞ − ⎜ ⎟ R1C1 ⎝ R6 R4R7 ⎠ R8 R6 R8 R2R3R7C1C2 1 R1C1 R2R8C2 R3R7C1 R7 R8 For given ai , bi , ki , C1, C2 and R8 : R1 = R2 = 1 a1C1 k1 a0 C2 R3 = 1 k1k 2 1 a0 C1 R4 = 1 1 1 k 2 a1b2 − b1 C1 R5 = k1 a0 b0C2 R6 = R8 b2 ωP = R8 R2R3R7C1C2 QP = ωP R1C1 R7 = k 2R8 EE315A ― HO #4 24 Sallen-Key LPF H (s ) = R.P. Sallen and E. L. Key “A Practical Method of Designing RC Active Filters.” IRE Trans. Circuit Theory, Vol. CT-2, pp. 74-85, 1955 ωP = G s s2 1+ + 2 ωPQP ωP 1 R1C1R2C2 • Single gain element • Poles only, no zeros • Sensitive to parasitic capacitances • Versions exist for HP, BP, … – http://en.wikipedia.org/wiki/Sallen_Key_filter QP = ωP 1 1 1− G + + R1C1 R2C1 R2C2 EE315A ― HO #4 B. Murmann 25 Tow-Thomas or Sallen-Key? • Suppose we now wanted to realize a biquad that has p pp q poles only y • Should we use a Tow-Thomas or Sallen-Key realization? • Clearly, from the perspective of complexity, we would probably want to go for Sallen-Key • Unfortunately, the Sallen-Key realization comes with disadvantages in terms of sensitivity to component variations • Let’s take a closer look… B. Murmann EE315A ― HO #4 26 Sensitivity • The sensitivity of any variable y to any parameter x is defined as (See e.g. Gray & Meyer, Section 4.2) Δy x ∂y ⎛ Δy / y ⎞ x y Sx = lim ⎜ = ⎟ = Δlim0 Δx → 0 ⎝ Δx / x ⎠ y x → Δx y ∂x • In order to relate fractional changes in y to fractional changes in x we can then write Δy y Δx ≅ Sx y x • Example y Sx = 10 • Δx = 2% x ⇒ Δy ≅ 20% y Common sense: Sensitivity is a first order approximation, accurate only for “small” errors small B. Murmann EE315A ― HO #4 27 Parameter Variations (1) • Discrete resistors and capacitors • Come in many different shapes and sizes and accuracies – E metal fil resistors, ~0.1% accurate, 5 E.g. t l film i t 0 1% t 5ppm/°C /°C – E.g. C0G dielectric capacitors, 2% accurate, very small temperature dependence B. Murmann EE315A ― HO #4 28 Parameter Variations (2) • Integrated resistors and capacitors Poly resistor • MIM Capacitor Important to distinguish between – Global process variations – Device-to-device mismatch EE315A ― HO #4 B. Murmann 29 Global Process Variations Wafer made yesterday All NMOS are “slow” All PMOS are “ “nominal” i l” All R are nominal All C are “fast” Parameter Wafer made today All NMOS are “fast” All PMOS are “f ” “fast” All R are nominal All C are “slow” “Slow” “Nominal” “Fast” 0.2V 0.3V 0.4V μCox (NMOS) C 240 μA/V2 A/V 300 μA/V2 A/V 360 μA/V2 A/V μCox (PMOS) 80 μA/V2 100 μA/V2 120 μA/V2 Rpoly 60Ω/□ 50Ω/□ 40Ω/□ Rnwell 1.4 kΩ/□ 1 kΩ/□ 0.6 kΩ/□ Vt CMIM B. Murmann 1.15 fF/μm2 1 EE315A ― HO #4 fF/μm2 0.85 fF/μm2 30 Device-to-Device Mismatch • Upon closer inspection, device parameters not only vary from lot-to-lot or wafer-to-wafer, but there are also differences between closely spaced, nominally id ti l d i b t l l d i ll identical devices on th the same chip – These differences are called mismatch Vt 1 − Vt 2 = ΔVt M1 M2 C1 R1 B. Murmann C2 R2 W⎞ ⎛ W⎞ ⎛ ⎜ μCox ⎟ − ⎜ μCox ⎟ = Δβ L ⎠1 ⎝ L ⎠2 ⎝ C1 − C2 = ΔC R1 − R2 = ΔR EE315A ― HO #4 31 Statistical Model • Experiments over the past decades have shown that device-todevice mismatch (ΔVt, ΔC, …) for properly laid out devices is typically “random” and well-described by a Gaussian distribution – With zero mean and a standard deviation that depends on the process and the size of the device • Empirically, the standard deviation of the mismatch between two closely spaced devices is modeled using the following expression σΔX = AX WL where W·L represents the area of the device, and X is the device parameter under consideration B. Murmann EE315A ― HO #4 32 Parameters for a Typical 0.18-μm Process Parameter Value AVt 5 mV-μm μ AΔβ/β 2 %-μm AΔC/C (MIM capacitor) 1 %-μm AΔR/R (Poly resistor) 3 %-μm EE315A ― HO #4 B. Murmann 33 Example • Example: MIM-capacitor with W=10μm, L=10μm (~100-200 fF) σΔC / C = 1% = 0.1% 100 3σΔC / C = 0.3% http://en.wikipedia.org/wiki/Image:Standard_deviation_diagram.svg B. Murmann EE315A ― HO #4 34 Sensitivity to Global Variations Tow-Thomas ωP = R8 1 ∝ R2R3R7C1C2 RC QP = ωP R1C1 ∝ 1 Sallen-Key ωP = QP = 1 R1C1R2C2 ∝ 1 RC ωP 1 1 1− G + + R1C1 R2C1 R2C2 • QP is independent of global variations in both realizations – Assuming all R and C use the same device structure structure, respectively • ∝1 ωP varies with the RC product of the components EE315A ― HO #4 B. Murmann 35 Sensitivity to Mismatch (Sallen-Key) ωP = QP = ω ω ω ω SR1P = SR2P = SC1P = SC2P = − 1 R1C1R2C2 ωP 1 1 1− G + + R1C1 R2C1 R2C2 Q Q SR1P = −SR2P = − R2C2 1 + QP 2 R1C1 Q Q SC1P = −SC2P = − ⎛ R1C2 R2C2 ⎞ 1 + QP ⎜ + ⎟ ⎜ RC 2 R1C1 ⎟ 2 1 ⎝ ⎠ Q SG P = QPG • 1 2 R2C1 R1C2 Sensitivity depends on QP and “component spread” i.e. the ratios of the resistors and capacitors, respectively B. Murmann EE315A ― HO #4 36 Example (1) • Want to design a Sallen-Key filter with QP=10 g y • Choice 1: All R and C are the same ⇒ G = 3 -(1/QP) = 2.9 – Very nice from the perspective of component spread, but bad for b d f sensitivity, e.g. iti it Q Q SR1P = −SR2P = − • 1 + QP = 9.5 2 Choice 2: Reduce sensitivity by accepting large component spread – Can show that G=1 is a good choice • See e.g. http://www.maxim-ic.com/appnotes.cfm/an_pk/738/ EE315A ― HO #4 B. Murmann 37 Example (2) • For G=1, we have QP = ωP 1 1 + R1C1 R2C1 and it follows that Q Q SR1P = −SR2P = − • R2 1 + =0 2 R1 + R2 R1 = R2 Unfortunately, Unfortunately in this case C1 2 = 4QP = 400 C2 • for for QP = 10 Bottom line: The Sallen-Key realization suffers from a strong tradeoff between sensitivity and component spread B. Murmann EE315A ― HO #4 38 Case Studies MAXIM APPLICATION NOTE 738 Minimizing Component-Variation Sensitivity in Single Op Amp Filters http://www.maxim-ic.com/appnotes.cfm/an_pk/738/ EE315A ― HO #4 B. Murmann 39 Sensitivity to Mismatch (Tow-Thomas) ωP = R8 R2R3R7C1C2 ω ω ω ω ωP ω SRP = SR P = SR P = −SR P = SC1 = SC P = − 2 3 3 8 2 1 2 Q SR1P = 1 QP = ωP R1C1 = R1 • R8C1 R2R3R7C2 Q Q Q Q Q Q SR2P = SR3P = SR7P = −SR8P = −SC1P = SC2P = − Constant sensitivities, independent of Q and component spread – Much nicer! B. Murmann EE315A ― HO #4 40 1 2 Conclusions • Biquads can be realized in numerous different ways • Implementation and component sizing have a big impact on p p g g p sensitivity to variations – Of course, we must avoid high-sensitivity circuits in practice • No theory f fi di a l N th for finding low-sensitivity architecture iti it hit t – Use proven circuits & check! • Tow-Thomas biquad q – Arbitrary poles and zeros, three amplifiers – Well-behaved sensitivities • Sallen-Key biquad – Only poles, one amplifier – Sensitivities trade off with component spread • Typically use G=1 and use this circuit only for “low Q” poles B. Murmann EE315A ― HO #4 41 ...
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This note was uploaded on 08/13/2009 for the course EE 315 taught by Professor Borismurmann during the Spring '09 term at Stanford.

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