Summary of Lecture 4

Transcription

1 EE47 Lecture 5 Filters Effect of integrator nonidealities on filter behavior Integrator quality factor and its influence on filter frequency characteristics (review for last lecture) Filter dynamic range limitations due to limited integrator linearity Measures of linearity: Harmonic distortion, intermodulation distortion, intercept point Effect of integrator component variations and mismatch on filter response Various integrator topologies utilized in monolithic filters Resistor based filters Transconductance (gm) based filters Switchedcapacitor filters ontinuoustime filter considerations Facts about monolithic Rs, gms, & s and its effect on integrated filter characteristics EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page Summary of Lecture 4 Ladder type RL filters converted to integrator based active filters All pole ladder type filters onvert RL ladder filters to integrator based form Example: 5 th order Butterworth filter High order ladder type filters incorporating zeros 7th order elliptic filter in the form of ladder RL with zeros Sensitivity to component mismatch ompare with cascade of biquads Doubly terminated L ladder filters Lowest sensitivity to component variations onvert to integrator based form utilizing SFG techniques Example: Differential high order filter implementation Effect of integrator nonidealities on continuoustime filter behavior Effect of integrator finite D gain & nondominant poles on filter frequency response EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page

2 Real Integrator NonIdealities Ideal Intg. log H ( s) Real Intg. log H ( s) a ω 0 ω P 0 a ω 0 PP3 ψ ψ 90 o 90 o ωo a H(s) H(s) s a s s ( s o )( p)( ω p3)... EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 3 Summary Effect of Integrator NonIdealities on Q Q intg. ideal Q intg. real o p i i a ω Amplifier D gain reduces the overall Q in the same manner as series/parallel resistance associated with passive elements Amplifier poles located above integrator unitygain frequency enhance the Q! If nondominant poles close to unitygain freq. Oscillation Depending on the location of unitygainfrequency, the two terms can cancel each other out! Overall quality factor of the integrator has to be much higher compared to the filter s highest pole Q EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 4

3 Effect of Integrator NonLinearities on Overall IntegratorBased Filter Performance Dynamic range of a filter is determined by the ratio of maximum signal output with acceptable performance over total noise Maximum signal handling capability of a filter is determined by the nonlinearities associated with its building blocks Integrator linearity function of opamp/r/ (or any other component used to build the integrator) linearity Linearity specifications for active filters typically given in terms of : Maximum allowable harmonic the output Maximum tolerable intermodulation distortion Intercept points & compression point referred to output or input EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 5 omponent Linearity versus Overall Filter Performance Ideal omponents Ideal D transfer characteristics: Perfectly linear output versus input tranfer function with no clipping Vout α for If Asin t Vout A t ( ω ) α sin( ω ) Vout Vout f f f f EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 6

4 omponent Linearity versus Overall Filter Performance SemiIdeal omponents Semiideal D transfer characteristics: Perfectly linear output versus input transfer function with clipping Vout α for Δ Δ Vout Δα for Δ Vout Δα for Δ If Asin ( ωt ) Vout α Asin ( ωt ) for Δ Δ lipped & distorted otherwise Vout Δ Δ Vout f f f f EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 7 Effect of omponent NonLinearities on Overall Filter Linearity Real omponents including NonLinearities Real D transfer characteristics: Both soft nonlinearities & hard (clipping) 3 Vout α α α3...for Δ Δ lipped otherwise If Asin t ( ω ) Vout f f Δ Δ Vout f? f EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 8

5 Effect of omponent NonLinearities on Overall Filter Linearity Real omponents including NonLinearities Typical real circuit D transfer characteristics: 3 Vout α α α3... If Asin ( ωt ) & A <Δ Then: Vout α Asin t ( ω ) α A sin( ωt) α3a sin( ωt) 3 α A α3 A ( t) ( cos( t) ) ( 3sin( t) sin( 3 )) or Vout α Asin ω ω ω ω t... 4 Vout Δ Δ Vout f f f f 3f f EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 9 Effect of omponent NonLinearities on Overall Filter Linearity Harmonic Distortion α ( ) A Vout α ( ( )) Asin ωt cos ωt 3 α3 A ( 3sin( ωt) sin( 3ωt ))... 4 nd amplitude harmonic distortion component HD amplitude fundamental rd amplitude3 harmonic distortion component HD3 amplitude fundamental α HD A, α α3 HD3 A 4 α EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 0

6 Example: Significance of Filter Harmonic Distortion in VoiceBand ODEs Voiceband ODE filter (ODE stands for coderdecoder, telephone circuitry includes ODEs with extensive amount of integrated active filters) Specifications includes limits associated with maximum allowable harmonic distortion at the output (< typically < % 40dB) ODE Filter including Output/Input transfer characteristic nonlinearity's Vout khz f khz 3kHZ f EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page Effect of omponent NonLinearities on Overall Filter Linearity Intermodulation Distortion D transfer characteristics including nonlinear terms, input sinusoidal waveforms: 3 Vout α α α3... If Asin ( ωt ) Asin ( ωt) Then Vout will have the following components: α αasin( ωt) α Asin( ωt) α α A sin ( ωt ) α A sin ( ωt) α AA sin( ωt ) sin ( ωt)... α A α A ( cos( ωt) ) ( cos( ωt) ) α AA cos( ( ω ω) t) cos( ( ω ω) t) α3 α3a sin ( ωt ) α3a sin ( ωt) 3α A A sin ωt sin ω t 3α A A sin ω t sin ω t 3A A ( ) ( ) 3 3 3α sin sin 4 ( ω ω ) t ( ω ω ) t ( ) ( ) EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page

7 Effect of omponent NonLinearities on Overall Filter Linearity Intermodulation Distortion Real D transfer characteristics, input sin waves: 3 Vout α α α3... If A sin ( ωt) A sn i ( ω t) Vout f f f f f f f f f f For f & f close in frequency omponents associated with (f f )& (f f ) are the closest to the fundamental signals on the frequency axis and thus most harmful EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 3 Effect of omponent NonLinearities on Overall Filter Linearity Intermodulation Distortion Intermodulation distortion is measured in terms of IM and IM3: Typically for input two sinusoids with equal amplitude ( A A A) nd amplitude IM component IM amplitude fundamental rd amplitude3 IM component IM 3 amplitude fundamental IM α 3α 5α A... IM A A α 4 α 8 α EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 4

8 Wireless ommunications Measure of Linearity db ompression Point Output Power (dbm) 0log( α ) db Vout α α α Input Power (dbm) P db EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 5 Wireless ommunications Measure of Linearity Third Order Intercept Point ω ω ω ω ω ω ω ω ω ω 3 Vout α α α3... 3rd IM 3 st 3α3 5α α 8 IIP3 Typically: IIP 3 P db 9.6 db OIP3 Output Power (dbm) 0log( α ) Most common measure of linearity for wireless circuits: OIP3 & IIP3, Third order output/input intercept point Fundamental 3 rd order IM Input Power (dbm) 3 3 0log α3 4 IIP3 EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 6

9 Example: Significance of Filter Intermodulation Distortion in Wireless Systems Typical wireless receiver architecture AG hannel Select Filters A/D Worst case signal scenario wrt linearity of the building blocks Two adjacent channels large compared to desired channel st Adjacent hannel Desired hannel f RX f n f n nd Adjacent hannel EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 7 Relative Signal Amplitude [db] Example: Significance of Filter Intermodulation Distortion in Wireless Systems st Adjacent hannel nd Adjacent hannel 60 RF Amp 30 Desired channel 30 not distinguishable 0 0 from intermod. omponent! f n f n Desired f RX f n f n f n f n f n f n hannel Adjacent channels can be as much as 60dB higher compared to the desired RX signal! Notice that in this example, 3 rd order intermodulation component associated with the two adjacent channel, falls on the desired channel signal! Relative Signal Amplitude [db] 60 EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 8

10 Filter Linearity Maximum signal handling capability is usually determined by the specifications wrt harmonic distortion and /or intermodulation distortion Distortion in a filter is a function of linearity of the components Example: In the above circuit linearity of the filter is mainly a function of linearity of the opamp voltage transfer characteristics EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 9 Various Types of Integrator Based Filter ontinuous Time Resistive element based OpampR OpampMOSFET OpampMOSFETR Transconductance (Gm) based Gm OpampGm Sampled Data Switchedcapacitor Integrator EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 0

11 ontinuoustime Resistive Element Type Integrators OpampR & OpampMOSFET & OpampMOSFETR R Vtune Vtune R OpampR OpampMOSFET OpampMOSFETR Vo ωo Ideal transfer function: where ωo s Req EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page ontinuoustime Transconductance Type Integrator Gm & OpampGm Gm Gm Gm Intg. Ideal transfer function: GmOTA Intg. Vo ωo Gm where ωo s EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page

12 Integrator Implementation Switchedapacitor φ φ I φ φ T/f clk s Vo f clk s for fsignal<< fclk V0 V dt in I ω s 0 fclk I Main advantage: ritical frequency function of ratio of caps & clock freq. ritical filter frequencies (e.g. LPF 3dB freq.) very accurate EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 3 Few Facts About Monolithic Rs & s & Gms Monolithic continuoustime filter critical frequency set by Rx or /Gm Absolute value of integrated Rs & s & Gms are quite variable Rs vary due to doping and etching nonuniformities ould vary by as much as ~0 to 40% due to process & temperature variations s vary because of oxide thickness variations and etching inaccuracies ould vary ~ 0 to5% Gms typically function of mobility, oxide thickness, current, device geometry ould vary > ~ 40% or more with process & temp. & supply voltage ontinuoustime filter critical frequency could vary by over 50% EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 4

13 Few Facts About Monolithic Rs & s While absolute value of monolithic Rs & s and gms are quite variable, with special attention paid to layout, & R & gms quite wellmatched Ratios very accurate and stable over processing, temperature, and time With special attention to layout (e.g. interleaving, use of dummy devices, commoncentroid geometries ): apacitor mismatch << 0.% Resistor mismatch < 0.% Gm mismatch < 0.5% EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 5 Impact of omponent Variations on Filter haracteristics Rs L 3 RL RL Filters Facts about RL filters ω 3dB determined by absolute value of Ls & s Shape of filter depends on ratios of normalized L & RL Norm Norm r R * ω 3dB L Norm R * L RL L L Norm r ω 3dB EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 6

14 Effect of Monolithic R & Variations on Filter haracteristics Filter shape (whether Elliptic with 0.dB Rpass or Butterworth..etc) is a function of ratio of normalized Ls & s in RL filters ritical frequency (e.g. ω 3dB ) function of absolute value of Ls xs Absolute value of integrated Rs & s & Gms are quite variable Ratios very accurate and stable over time and temperature What is the effect of onchip component variations on monolithic filter frequency characteristics? EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 7 Impact of Process Variations on Filter haracteristics Rs L 3 RL * R V Rs in sτ sτ sτ 3 * R R L RL Filters Integrator Based Filters RL * Norm τ.r ω 3dB L RL L Norm τ R * ω 3dB τ Norm τ L Norm EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 8

15 Impact of Process Variations on Filter haracteristics I R R R3 I I3 int g int g τ τ Norm Norm L R R In Rn Rn R3 R3 τ τ int g int g τ τ int g int g Norm I.R ω 3dB Norm L.R I ω 3dB Norm.R I Norm I.R L Variation in absolute value of integrated Rs & s change in critical freq. (ω 3dB ) Since Ratios of Rs & s very accurate ontinuoustime monolithic filters retain their shape due to good component matching even with variability in absolute component values EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 9 Example: LPF Worst ase orner Frequency Variations Nominal Bandwidth Detailed passband (note shape is wellretained) Worst case bandwidth variation While absolute value of onchip R (gm) timeconstants could vary by as much as 00% (process & temp.) With proper precautions, excellent component matching can be achieved: Wellpreserved relative amplitude & phase vs freq. charactersitics Need to only adjust (tune) continuoustime filter critical frequencies EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 30

16 Tunable OpampR Filters Example st order OpampR filter is designed to have a corner frequency of.6mhz Assuming process variations of: varies by 0% R varies by 5% Build the filter in such a way that the corner frequency can be adjusted postmanufacturing. R0KΩ R0KΩ 0pF Nominal R & values for.6mhz corner frequency EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 3 Filter orner Frequency Variations Assuming expected process variations of: Maximum variations by 0% nom0pf min9pf, maxpf Maximum R variations by 5% Rnom0K Rmin7.5K, Rmax.5K orner frequency ranges from.357mhz to.57mhz orner frequency varies by 48% & 7% EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 3

17 Variable Resistor or apacitor In order to make provisions for filter to be tunable either R or should be adjustable (this example adjustable R) Monolithic Rs can only be made adjustable in discrete steps (not continuous) max Rnom f max.48 Rnom fnom max Rnom 4.8kΩ min Rnom f min Rnom fnom min Rnom 7.k Ω 0.7 D D D0 R R R3 R4 Variable Resister MOS xtors act as switches Dn high switch acts as short circuit Dn low switch open circuit EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 33 Tunable Resistor Maximum variations by 0% min9pf, maxpf Maximum R variations by 5% Rmin7.5K, Rmax.5K orner frequency varies by 48% & 7.% Assuming control signal has n 3bit (0 or ) for adjustment RR34R4 min R Rnom 7.kΩ n ( max min R nom nom ) ( ) R R 4.8k 7.k 4 n kΩ n min ( max R nom nom ) ( ) 3 R R 4.8k 7.k n 7.7kΩ n 3 min ( max R nom nom ) ( ) 4 R R 4.8k 7.k n 7.08kΩ Tuning resolution.08k/0k 0% D D D0 R R R3 R4 Variable Resister MOS xtors act as switches EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 34

18 Tunable OpampR Filter D D D0 Rnom 7.K D D D K K K R R R3 R4 Post manufacturing: Set all Dx to 00 (mid point) Measure 3dB frequency R R R3 R4 If frequency too high decrement D to D If frequency too low increment D to D If frequency within 0% of the desired corner frequency stop else For higher order filters, all filter integrators tuned simultaneously EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 35 Tunable OpampR Filters Summary Tunable Opamp R Integrator Program s and/or Rs to freq. tune the filter All filter integrators tuned simultaneously Tuning in discrete steps & not continuous Tuning resolution limited Switch parasitic & series R can affect the freq. response of the filter EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 36

19 Example: Tunable LowPass OpampR Filter Adjustable apacitors EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 37 Opamp R Filters Advantages Since resistors are quite linear, linearity only a function of opamp linearity good linearity Disadvantages Opamps have to drive resistive load, low output impedance is required High power consumption ontinuous tuning not possibletuning only in discrete steps Tuning requires programmable Rs and/or s EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 38

20 Integrator Implementation OpampR & OpampMOSFET & OpampMOSFETR R Vtune Vtune R OpampR OpampMOSFET OpampMOSFETR Vo ωo where ωo s Req EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 39 Use of MOSFETs as Variable Resistors R OpampR R replaced by MOSFET Operating in triode mode ontinuously variable resistor: Vtune OpampMOSFET I D Triode region V GS MOSFET IV characteristic: Nonlinear R V DS EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 40

21 Opamp MOSFET Integrator SingleEnded Integrator W V ID μ ox ds ( Vgs V L th ) Vds W V i I μ ox ( Vgs V D th ) V L i ID W G μ ox ( Vgs Vth Vi) Vi L By varying VG effective admittance is tuned Tunable integrator time constant VG I D Tunable by varying VG: Problem: Singleended MOSFET Integrator Effective R nonlinear Note that the nonlinearity is mainly nd order type EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 4 Use of MOSFETs as Resistors Differential Integrator W V ds ID μ ox Vgs Vth V L ds W Vi V I i D μ ox Vgs V L th 4 W Vi V I i D μ ox Vgs V L th 4 W I D I D μ ox ( V gs V th ) V L i ( ID ID ) W G μ ox Vgs Vth Vi L ( ) Vi/ Vi/ I D I D M VG M ut Nonlinear term is of even order & cancelled! Admittance independent of Vi OpampMOSFET Problem: Threshold voltage dependence EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 4

22 Use of MOSFET as Resistor Issues MOS xtor operating in triode region ross section view Distributed channel resistance & gate capacitance Distributed nature of gate capacitance & channel resistance results in infinite no. of highfrequency poles: Excess the unitygain frequency of the integrator Enhanced integrator Q Enhanced filter Q, Peaking in the filter passband EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 43 Use of MOSFET as Resistor Issues MOS xtor operating in triode region ross section view Distributed channel resistance & gate capacitance Tradeoffs affecting the choice of device channel length: Filter performance mandates wellmatched MOSFETs long channel devices desirable Excess phase increases with L Q enhancement and potential for oscillation! Tradeoff between device matching and integrator Q This type of filter limited to low frequencies EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 44

23 Suitable for low frequency applications Issues with linearity Linearity achieved ~40 50dB Needs tuning Example: Opamp MOSFET Filter 5 th Order Elliptic MOSFET LPF with 4kHz Bandwidth Ref: Y. Tsividis, M.Banu, and J. Khoury, ontinuoustime MOSFET Filters in VLSI, IEEE Journal of Solid State ircuits Vol. S, No. Feb. 986, pp. 530 EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 45 Improved MOSFET Integrator W V ds ID μ ox Vgs Vth V L ds W V V I i i D μ ox Vgs V L th 4 W Vi V I i D3 μ ox Vgs3 Vth Vi/ L 4 IX ID ID3 W Vi V μ i ox Vgs V L gs3 W Vi V I i Vi/ X μ ox Vgs3 V L gs W I X I X μ ox ( V gs V gs3 ) V L i ( IX IX) G μ W ox ( V gs V gs3 ) Vi L No threshold voltage dependence V G V G3 M I D I D I D3 I D4 M4 I X M3 I X M M,,3,4 equal W/L ut Linearity achieved in the order of 5070dB Ref: Z. zarnul, Modification of the BanuTsividis ontinuoustime Integrator Structure, IEEE Transactions on ircuits and Systems, Vol. AS33, No. 7, pp. 7476, July 986. EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 46

24 RMOSFET Integrator V G V G Vi/ Vi/ R R M M M4 M3 ut Improvement over MOSFET by adding resistor in series with MOSFET Voltage drop primarily across fixed resistor small MOSFET Vds improved linearity & reduced tuning range Generally low frequency applications Ref: UK Moon, and BS Song, Design of a LowDistortion khz Fifth Order Bessel Filter, IEEE Journal of Solid State ircuits, Vol. 8, No., pp. 5464, Dec EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 47 RMOSFET Lossy Integrator R Vi/ Vi/ R R V G M M M4 V G M3 ut Negative feedback around the nonlinear MOSFETs improves linearity but compromises frequency response accuracy Ref: UK Moon, and BS Song, Design of a LowDistortion khz Fifth Order Bessel Filter, IEEE Journal of Solid State ircuits, Vol. 8, No., pp. 5464, Dec EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 48 R

25 Example: Opamp MOSFETR Filter 5 th Order Bessel MOSFETR LPF khz bandwidth THD 90dB for 4Vpp, khz input signal Suitable for low frequency, low Q applications Significant improvement in linearity compared to MOSFET Needs tuning Ref: UK Moon, and BS Song, Design of a LowDistortion khz Fifth Order Bessel Filter, IEEE Journal of Solid State ircuits, Vol. 8, No., pp. 5464, Dec EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 49 Operational Amplifiers (Opamps) versus Operational Transconductance Amplifiers (OTA) Opamp Voltage controlled voltage source OTA Voltage controlled current source Output in the form of voltage Low output impedance an drive Rloads Good for R filters, OK for S filters Extra buffer adds complexity, power dissipation Output in the form of current High output impedance In the context of filter design called gmcells annot drive Rloads Good for S & gm filters Typically, less complex compared to opamp higher freq. potential Typically lower power EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 50

26 Integrator Implementation Transconductance & OpampTransconductance Gm Gm Gm Intg. GmOTA Intg. Vo ωo Gm where ωo s EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 5 Gm Filters Simplest Form of MOS Gm Integrator Transconductance element formed by the sourcecoupled pair M & M All MOSFETs operating in saturation region urrent in M& M can be varied by changing V control Transconductance of M& M varied through V control int g M M M0 V control Ref: H. Khorramabadi and P.R. Gray, High Frequency MOS continuoustime filters, IEEE Journal of SolidState ircuits, Vol.S9, No. 6, pp , Dec EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 5

27 Simplest Form of MOS Gm Integrator A Half ircuit int g int g M M M0V V control control int g int g M M V M0 control M intg A half circuit EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 53 Gm Filters Simplest Form of MOS Gm Integrator Use ac half circuit & small signal model to derive transfer function: M, Vo gm int gs M, Vo g m int g s Vo ω o s M, gm ωo int g GS M gm intg A half circuit Small signal model intg EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 54

28 Gm Filters Simplest Form of MOS Gm Integrator MOSFET in saturation region: μ W I ox d ( Vgs Vth ) L Gm is given by: M&M Id W gm μ ox Vgs Vth Vgs L I d V gs V th ( ) / W μ ox I L d ( ) Id varied via Vcontrol gm tunable via Vcontrol M int g M M0 V control EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 55 Gm Filters nd Order Gm Filter Use the Gmcell to build a nd order bandpass filter M int g M M0 V control EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 56

29 nd Order Bandpass Filter Iin V R R V L L V I I R L I V ' V * R R V * sr ' V 3 * R sl τ τ * * R L R * R R sτ sτ EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 57 nd Order IntegratorBased Bandpass Filter VBP s ττ s βτs τ * * R L R * β R R ω τ 0 τ τ Q β τ τ τ L Q V BP sτ sτ From matching pointof viewdesirable: Q R ω0 * R τ τ τ EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 58

30 nd Order IntegratorBased Bandpass Filter V BP First implement this part With transfer function: Q sτ sτ V0 s ω0 Q EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 59 Terminated Gm Integrator int g M3 M M0 M4 M M V control M M3 A half circuit intg EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 60

31 Terminated Gm Integrator M M3 intg GS g M m M 3 g m intg A half circuit Small signal model Vo V M 3 in int g g s m M M gm gm V0 ompare to: s ω0 Q EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 6 Terminated Gm Integrator Q V BP sτ sτ GS g M m M 3 g m intg V0 s ω0 Q Small signal model Vo V M 3 in s int g g m M M gm gm M M gm g ω m 0 & Q M3 int g gm Question: How to define Q accurately? EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 6

32 Terminated Gm Integrator / M WM M gm μ ox I d LM / M3 WM3 M3 gm μ ox I L d M3 Let us assume equal channel lengths for M, M3 then: M / M g m I d W M M3 M3 g W m I d M3 int g M3 M4 M M M M0 VVcontrol EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 63 Terminated Gm Integrator Note that: M M0 I d I d M3 M I d I d Assuming equal channel lengths for M0, M: int g M0 I M3 M4 d W M0 M M M I d W M / M M W gm M0 WM M0 VVcontrol M3 W g m W M M3 EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 64

33 nd Order Gm Filter Simple design Tunable Q function of device ratios: g Q g M, m M 3,4 m EES 47 Lecture 5: IntegratorBased Filters 009 H.K. Page 65