HO6_315aSP09_high_order_2

HO6_315aSP09_high_order_2 - Realization of High Order...

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Unformatted text preview: Realization of High Order Filters High-Order Part II Boris Murmann Stanford University [email protected] Copyright © 2009 by Boris Murmann EE315A ― HO #6 B. Murmann 1 Sensitivity Problem with Cascaded Biquads x x x In x Out x x x x [Moon] x x • Passband response is sensitive to shifts in the pole positions – Especially for high Q • Typically, integrated continuous time filters use biquads to realize filters only up to ~5th order B. Murmann EE315A ― HO #6 2 Conceptual View of a Biquad Cascade • Individual sections are actively decoupled – Variations in individual components affect only one pair of poles (and/or zeros) • Ideally, we would like all the poles (and zeros) to “move together” – This would at least preserve the “shape” of the filter response EE315A ― HO #6 B. Murmann 3 Doubly Terminated LC Ladder Filters x x x x x • The passband response of ladder filters is much less sensitive to component variations when compared to biquad cascade – Poles “tend” to move together • For a sensitivity analysis see e g analysis, e.g. – G. C. Temes and H. J. Orchard, “First order sensitivity and worst-case analysis of doubly terminated reactance two-ports,” IEEE Trans. Circuit Theory, 20 (5), pp. 567–571, 1973. B. Murmann EE315A ― HO #6 4 Analysis Example (1) i 4 = v outY4 = i3 v 2 = i3Z3 + v out i 2 = v 2Y2 i1 = i 2 + i 3 v in = i1Z1 + v 2 = (v 2Y2 + v outY4 ) Z1 + v 2 = ([Y4Z3v out + v out ]Y2 + voutY4 ) Z1 + v outY4Z3 + v out v out 1 = v in ([Y4Z3 + 1]Y2 + Y4 ) Z1 + Y4Z3 + 1 = 1 Y4Z3Y2Z1 + Y4Z3 + Y4Z1 + Y2Z1 + 1 EE315A ― HO #6 B. Murmann 5 Analysis Example (2) • E.g. for Z1 = R1 Y2 = sC2 Z3 = sL3 Y4 = sC4 it follows that v out 1 = 3 2 v in s C4L3C2R1 + s C4L3 + s (C4R1 + C2R1 ) + 1 • A third order lowpass filter • Zeros can be realized by utilizing parallel and series combinations of inductors and capacitors p • Analysis is doable − But very tedious! B. Murmann EE315A ― HO #6 6 LC Ladder Synthesis • Filter tables – A. Zwerev, Handbook of filter synthesis, Wiley, 1967 – R. Saal, Handbook of filter synthesis, AEG-Telefunken, 1979 – A. B. Williams and F. J. Taylor, Electronic filter design, 3rd edition, McGraw-Hill, 1995 • CAD tools – http://www.circuitsage.com/filter.html • Comprehensive list of available tools – http://tonnesoftware.com/elsie.html • Free version of Elsie supports ladder synthesis up to 7th order – http://www.nuhertz.com/download.html • FilterFree – up to 3rd order • FilterSolutions – $$$ – Agilent ADS EE315A ― HO #6 B. Murmann 7 Butterworth Filter Table • Denormalization Li ,den = Li Ci ,den = Ci • R ω−3dB 1 ω−3dB ⋅ R R is the desired value of th source l f the and termination resistor [Schaumann] B. Murmann EE315A ― HO #6 8 5th Order Elliptic Filter Table (1) ⎛ ωp ⎞ ⎟ ⎝ ωs ⎠ θ = sin−1 ⎜ 0.17dB passband ripple [Williams & Taylor] Table 11-56 T bl 11 56 B. Murmann EE315A ― HO #6 9 5th Order Elliptic Filter Table (2) B. Murmann EE315A ― HO #6 10 Back to Our Design Example • Channel select filters (CSF) – 640 kHz passband, p , lowpass – 0.5 dB passband ripple – > 40 dB stopband attenuation at 900 kHz • 5th order elliptical filter Lim et al., “A Fully Integrated Direct-Conversion Receiver for CDMA and GPS Applications,” IEEE JSSC, Nov. 2006 Applications, EE315A ― HO #6 B. Murmann 11 Synthesis Result (Using Elsie) L2 L4 C2 C1 • C4 C3 C5 Termination resistors arbitrarily set to 10kΩ B. Murmann EE315A ― HO #6 12 Spice Simulation Result -6 Magnitude [dB] Magnitud [dB] de -10 -20 -30 -6.5 -7 -7.5 -8 -40 -8 5 8.5 -50 4 10 • 5 10 10 Frequency [Hz] 6 10 7 10 4 5 10 Frequency [Hz] 6 dB passband attenuation due to resistive termination – Easy to change to 0dB in an active realization EE315A ― HO #6 B. Murmann 13 20% Variation in L2 +20% -20% -5 5 5.5 Mag gnitude [dB] Mag gnitude [dB] -5 +20% -20% -10 -20 -30 30 -6 -6.5 -7 -40 -7.5 -50 4 50 10 • 5 10 10 Frequency [Hz] 6 10 7 10 4 5 10 Frequency [Hz] 10 Only a very small change in the passband response; moderate degradation in the stopband – Smaller (i.e. more realistic) variations than 20% can be easily handled through overdesign B. Murmann EE315A ― HO #6 14 6 State-Space Description for C1 C2 in V1 = + V1 - I2 C4 + V3 - I4 + V5 - out I1 1 ⎛ Vin − V1 ⎞ = ⎜ R − I2 + [V3 − V1 ] sC2 ⎟ sC1 sC1 ⎝ ⎠ ⎛ C ⎞ ⎛ V − V1 C ⎞ 1 V1 ⎜ 1 + 2 ⎟ = ⎜ in − I2 ⎟ + V3 2 C1 ⎠ ⎝ R C1 ⎠ sC1 ⎝ V1 = − ⎛ Vin V1 V2 ⎞ 1 + + − V3sC2 ⎟ ⎜− s (C1 + C2 ) ⎝ R R Rx 2 ⎠ EE315A ― HO #6 B. Murmann 15 Implementation of C1 Integrator V1 = − B. Murmann ⎛ Vin V1 V2 ⎞ 1 + + − V3sC2 ⎟ ⎜− s (C1 + C2 ) ⎝ R R Rx 2 ⎠ EE315A ― HO #6 16 Implementation of L2 Integrator C2 C4 I2 = in + V1 - I2 + V3 - I4 + V5 - out 1 (V1 − V3 ) sL2 V2 = I2Rx 2 = − 2 V ⎞ Rx 2 ⎛ V1 + 3 ⎟ ⎜− sL2 ⎝ Rx1 R x 2 ⎠ EE315A ― HO #6 B. Murmann 17 Remaining Integrators V3 = − ⎛ V2 ⎞ 1 V + 4 − V1sC2 − V5sC4 ⎟ ⎜− s (C2 + C3 + C4 ) ⎝ Rx 4 Rx 4 ⎠ V4 = I4Rx 4 = − Vout = V5 = − B. Murmann 2 V ⎞ R x 4 ⎛ V3 + 5 ⎟ ⎜− sL4 ⎝ R x 4 R x 4 ⎠ ⎛ V5 V4 ⎞ 1 − V3sC4 ⎟ ⎜ − s (C4 + C5 ) ⎝ R Rx 4 ⎠ EE315A ― HO #6 18 - C4+C5 R L4/Rx42 R - C2+C3+C4 - C1+C2 R - - L2/Rx22 R Complete Realization EE315A ― HO #6 B. Murmann 19 Signal Inversion -1 Symbol Realization in a single ended circuit single-ended (need only one shared circuit per state) - V1 1p V2 2p V1m V2m Realization in a differential circuit • In a first-cut (single-ended) simulation, signal inversion can also be achieved using negative resistors and capacitors B. Murmann EE315A ― HO #6 20 Simulation Setup • AC analysis with 1V applied at the input • Amplifiers are ideal ideal, with an open loop gain of 106 • Set Rx2=Rx4=R=10kΩ – Somewhat arbitrary at this point EE315A ― HO #6 B. Murmann 21 Frequency Response 1.6 Magnitude 1.4 V4 1.2 12 |V1|max = 0.8505 V |V2|max = 1.5585 V |V3|max = 0.9039 V |V4|max = 1.7072 V 1 7072 |Vout|max = 0.5000 V 1 V2 0.8 0.6 06 0.4 0.2 0 V1 10 V3 Vout 5 10 6 Frequency [Hz] B. Murmann EE315A ― HO #6 22 Node Voltage Scaling • To scale the peak output voltage of an integrator by a factor of k, scale all resistors and capacitors connected to the input and output node as shown b l t t d h below • Feedback R and C remain unchanged – Will be scaled together with all other components to adjust g p j the thermal noise (more later) EE315A ― HO #6 B. Murmann 23 Frequency Response After 0dB Scaling 1 Magnitude e 0.8 0.6 0.4 0.2 0 10 5 10 6 Frequency [Hz] B. Murmann EE315A ― HO #6 24 Component Values After 0dB Scaling • % zero C [F] c13 = -6.4665e-012 c31 = -7.3035e-012 c35 = -3.6888e-011 c53 = -1 1287e-011 1.1287e 011 Capacitors – Cmin = 6.47 pF – Cmax = 48.7 pF • % coupling R [Ω] rin = -8.5052e+003 r12 = -1.8324e+004 r21 = 5.4573e+003 r23 = -5.7997e+003 r32 = 1.7242e+004 r34 = -1.8888e+004 r43 = 5.2944e+003 r45 = -2.9287e+003 r54 = 3.4145e+004 Resistors – Rmin = 2.93 kΩ – Rmax = 34.1 kΩ • % feedback R [Ω] and C [F] ci1 = 4.3711e-011 ri1 = 10000 ci2 = 2.5416e-011 i2 2 5416 011 ci3 = 7.4108e-011 ci4 = 1.7567e-011 ci5 = 4.8692e-011 ri5 = 10000 Component spread ~10 10 – Manageable in practice – May be able to reduce spread by scaling integration capacitors capacitors, subject to noise constraints • A very complex optimization problem! EE315A ― HO #6 B. Murmann 25 Schematic r12 r32 ci2 ci1 ri1 c13 B. Murmann r21 r54 ci4 ci3 c31 rin r34 c35 ci5 ri5 c53 r23 EE315A ― HO #6 r43 r45 26 20% Variation in Ci2 • Only small passband variations despite large component variation • Active realization of ladder retains low sensitivity of passive prototype • More analysis is needed to determine the actual precision requirements for all components – E through a Monte Carlo simulation E.g. th h M t C l i l ti EE315A ― HO #6 B. Murmann 27 Summary • Higher-order filter realization g – Cascade of biquads • High sensitivity often problematic for order ≥ 5 – Ladder filters • Based on LC prototypes • Low sensitivity • Active RC simulation retains low sensitivity B. Murmann EE315A ― HO #6 28 ...
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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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