EECS 247 Lecture 8: Filters: Gm-C & S.C.© 2007 H.K. Page 1EE247 Lecture 8• Continuous-time filters continued– Various Gm-C filter implementations– Comparison of continuous-time filter topologies• Switched-Capacitor Filters– “Analog” sampled-data filters:• Continuous amplitude• Quantized time– Applications:• First commercial product: Intel 2912 voice-band CODEC chip, 1979• Oversampled A/D and D/A converters• Stand-alone filtersE.g. National Semiconductor LMF100 (x2 biquads)EECS 247 Lecture 8: Filters: Gm-C & S.C.© 2007 H.K. Page 2Summary Last Lecture• Automatic on-chip filter tuning (continued from previous lecture)– Continuous tuning• Reference integrator locked to a reference frequency• DC tuning of resistive timing element– Periodic digitally assisted tuning• Systems where filter is followed by ADC & DSP, existing hardwarecan be used to periodically update filter freq. response• Continuous-time filters– Highpass filters- 1storderÆ integrator in the feedback path– Bandpass filters• Cascade of LP and HP for Qfilter<5• Direct implementation for narrow-band filter via LP to BP transformationEECS 247 Lecture 8: Filters: Gm-C & S.C.© 2007 H.K. Page 3Simplest Form of CMOS Gm-CellNonidealities• DC gain (integrator Q)• Where a denotes DC gain & θis related to channel length modulation by: • Seems no extra poles!()M1,2mM1,20loadM1,2gagg2LaVVgs thLθθλ=+=−=Small Signal Differential Mode Half-CircuitEECS 247 Lecture 8: Filters: Gm-C & S.C.© 2007 H.K. Page 4CMOS Gm-Cell High-Frequency Poles• Distributed nature of gate capacitance & channel resistance results in infinite no. of high-frequency polesCross section view of a MOS transistor operating in saturationDistributed channel resistance & gate capacitanceEECS 247 Lecture 8: Filters: Gm-C & S.C.© 2007 H.K. Page 5CMOS Gm-Cell High-Frequency Poles• Distributed nature of gate capacitance & channel resistance results in an effective pole at 2.5 times input device cut-off frequencyHigh frequency behavior of an MOS transistor operating in saturation region()M1,2M1,2effective2i2ieffectivet2M1,2M1,2mt21P1PP2.5VVgs thg3C2/3 WL 2Loxμωω∞=≈≈−==∑EECS 247 Lecture 8: Filters: Gm-C & S.C.© 2007 H.K. Page 6Simple Gm-Cell Quality Factor()M1,2effective22VVgs th15P4Lμ−=()M1,22LaVVgs thθ=−• Note that phase lead associated with DC gain is inversely prop. to L• Phase lag due to high-freq. poles directly prop. to LÆ For a given ωο there exists an optimum L which cancel the lead/lag phase error resulting in high integrator Q()()i11opi22M1,2oM1,2intg.1realVVgs thL14intg.2L 15VVgs thQaQωθωμ∞=≈−−≈−−∑EECS 247 Lecture 8: Filters: Gm-C & S.C.© 2007 H.K. Page 7Simple Gm-Cell Channel Length for Optimum Integrator Quality Factor()1/32M1,2oVVgs th.15opt.4Lθμω⎡⎤−⎢⎥≈⎢⎥⎢⎥⎣⎦• Optimum channel length computed based on process parameters (could vary from process to process)EECS 247 Lecture 8: Filters: Gm-C & S.C.© 2007 H.K. Page 8Source-Coupled Pair CMOS Gm-Cell Transconductance()()()1/ 22iidssM1,2M1,2iM1,M 2dmiM1,2diivv1II1VVVV4gs thgs thvINote:For small gVVvgs thINote: As v increases or the veffective transconductance decreases⎧⎫Δ⎡⎤Δ⎡⎤⎪⎪Δ=⎢⎥−⎢⎥⎨⎬−−⎢⎥⎢⎥⎪⎪⎣⎦⎣⎦⎩⎭Δ⎡⎤Δ→=⎢⎥−Δ⎢⎥⎣⎦ΔΔΔFor a source-coupled pair the differential output current (ΔId)as a function of the input voltage(Δvi):ii1i2dd1d2vV VIIIΔ= −Δ= −EECS 247 Lecture 8: Filters: Gm-C & S.C.© 2007 H.K. Page 9Source-Coupled Pair CMOS Gm-Cell LinearityIdeal Gm=gm• Large signal Gmdrops as input voltage increasesÆ Gives rise to nonlinearityEECS 247 Lecture 8: Filters: Gm-C & S.C.© 2007 H.K. Page 10Measure of Linearityω1ω13ω1ωω2ω1−ω22ω2−ω1Vin Voutω1ωω2ω1ωω2Vin Vout2312 32313243511.............3..31......43.325......48Vout Vin Vin Vinamplitude rd harmonicdist compHDamplitude fundamentalVinamplitude rdorder IM compIMamplitude fundamentalVin Vinαα ααααααα=+ + +==+==+ +EECS 247 Lecture 8: Filters: Gm-C & S.C.© 2007 H.K. Page 11Source-Coupled Pair Gm-Cell Linearity()()()()()1/22iidssM1,2M1,223d1 i2 i 3 iss12M1,2ss343M1,2ss5vv1II (1)1VVVV4gs thgs thI a v a v a v .............Series expansion used in (1)Ia&a0VVgs thIa&a08V Vgs thIa128 V Vgs th⎧⎫Δ⎡⎤Δ⎡⎤⎪⎪Δ=⎢⎥−⎢⎥⎨⎬−−⎢⎥⎢⎥⎪⎪⎣⎦⎣⎦⎩⎭Δ=×Δ+×Δ+×Δ+==−=− =−=−−65M1,2&a0=EECS 247 Lecture 8: Filters: Gm-C & S.C.© 2007 H.K. Page 12Linearity of the Source-Coupled Pair CMOS Gm-Cell• Key point: Max. signal handling capability function of gate-overdrive voltage() ()()()2435ii111324iiGS th GS thimax GS thrms3GSth in3a 2 5aˆˆIM3 v v ............4a 8aSubstituting for a ,a ,....ˆˆvv325IM3 ............32 1024VV VV2ˆv4VV IM33ˆIM 1% & V V 1V V 230mV≈+⎛⎞⎛⎞≈+⎜⎟⎜⎟−−⎝⎠⎝⎠≈−××=−=⇒ ≈EECS 247 Lecture 8: Filters: Gm-C & S.C.© 2007 H.K. Page 13Simplest Form of CMOS Gm CellDisadvantages()()()sincethen23GSthM1,2moint goIM V Vg2CWVVCggsthmoxLVVgs thωμω−∝−=×−=−∝•Max. signal handling capability function of gate-overdrive•Critical freq. is also a function of gate-overdriveÆ Filter tuning affects max. signal handling capability!EECS 247 Lecture 8: Filters: Gm-C & S.C.© 2007 H.K. Page 14Simplest Form of CMOS Gm CellRemoving Dependence of Maximum Signal Handling Capability on TuningÆ Dynamic range dependence on tuning removed (to 1storder)Ref: R.Castello ,I.Bietti, F. Svelto , “High-Frequency Analog Filters in Deep Submicron Technology , “International Solid State Circuits Conference, pp 74-75, 1999.• Can overcome problem of max. signal handling capability being a function of tuning by providing tuning through :– Coarse tuning via switching in/out binary-weighted cross-coupled pairsÆ Try to keep gate overdrive voltage constant– Fine tuning through varying current sourcesEECS 247 Lecture 8:
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