EECS 6.012 Spring 1998Lecture 10 I. MOSFET Circuit ModelsA. Large Signal Model - NMOS • Cutoff : ( VGS ≤ VTn )-----> ID = 0 • Triode : ( VGS ≥ VTn and VDS ≤ VGS - VTn )• CLM term added to ensure continuous curve for ID vs. VDS • Saturation : ( VGS ≥ VTn and VDS ≥ VGS - VTn ). B. Backgate Effect • The threshold voltage is a function of the bulk-to-source voltage • where V TOn is the threshold voltage with V BS = 0 • γ n is the backgate effect parameter IDµnCoxWL⁄()VGSVTn– VDS2⁄– VDS1 λnVDS+=ID12⁄()µnCoxWL⁄()VGSVTn–21λnVDS+=VTnVTOnγnV–BS2– φp2– φp–+=γn2qεsNaCox⁄=EECS 6.012 Spring 1998Lecture 10 II. MOSFET Small-Signal ModelA. Small Signal Modelling Concepts • Find an equivalent circuit which relates the incremental changes in i D , v GS , v DS , etc. • Since the changes are small, the small-signal equivalent circuit has linear elements only (e.g., capacitors, resistors, controlled sources)• Mathamatically we perform a Taylor expansion around the DC operating point (also called the quiescent point or Q point) defined by the DC voltages Q(V GS , V DS , V BS ):• The total drain current in saturation: i D = (1/2) µ n C ox ( W/L ) ( v GS - V Tn ) 2 (1 + λ n v DS ) = iD(vGS, vDS, vBS)where vGS = VGS + vgs , iD = ID + id • We want to find id = (?) vgsIf the small-signal voltage is really “small,” then we can neglect everything past the linear term --where the partial derivative is defined as the transconductance, gm.iDIDvGS∂∂iDQvgs()12---vGS22∂∂ iDQvgs()2…++ +=iDIDvGS∂∂iDQvgs()+ IDgmvgs+==EECS 6.012 Spring 1998Lecture 10B. Transconductance• The small-signal drain current due to vgs is therefore given byid = gm vgs.DSG+_BVDS = 3 V+_VGS = 3 V+_vgsiD = ID + id1100200300400iD(µA)vDS (V) VGS + vgs QID + id VDS5432IDidid = gmvgsVGSEECS 6.012 Spring 1998Lecture 10C. Quantifying Transconductance • Evaluating the partial derivative:• We neglect the effect of CLM when calculating the transconductance so that gm in terms of VGS becomes• In many circuits we want an expression for gm in terms of the DC drain current• For typical values (W/L) = 10, ID = 100 µA, and µnCox = 50 µAV-2 we find that gm = 320 µAV-1 = 0.32 mS• The circuit which expresses id = gm vgs gmµnCoxWL-----VGSVTn–()1λnVDS+()=gmµnCoxWL-----VGSVTn–()=gm2µnCoxWL-----ID=gmvgsgatesourcedrain+_vgsidEECS 6.012 Spring 1998Lecture 10D. Output Conductance• The change in drain current due to an incremental change in the drain-source voltage is:• The output resistance is the inverse of the output conductance• The small-signal circuit model with ro added looks like:iD(µA)ID + id vdsVDSVDS + vds VDS (V) IDidid = govdsVGS, VBS1100200300400Q5432goiD∂vDS∂------------Q12---µnCoxWL-----VGSVT–()2λnλnID≅==ro1go-----1λnID------------==gmvgsrogatesourcedrain+_vgsid+_vdsid = gm vgs + (1/ro)vdsEECS 6.012 Spring 1998Lecture 10E. Backgate Transconductance• The change in drain current due to an incremental change in the backgate bias is found using the by the chain rule:.iD(µA)ID + id VDSVDS (V) IDidid = gmbvbs VGS, VBS + vbs VGS, VBS1100200300400Q5432gmbiD∂vBS∂------------QiD∂VTn∂------------QVTn∂vBS∂------------Q==iD∂VTn∂------------Qµ–nCoxWL-----VGSVTn–()1λnVDS+()g–m≈=gmbgm–()VTn∂vBS∂------------Qgm–()γn–22–φpVBS–------------------------------------γngm22–φpVBS–------------------------------------== =EECS 6.012 Spring 1998Lecture 10E. Backgate Transconductance (con’t)• The ratio of the backgate transconductance gmb to the “front-gate” transconductance gm to is:• where Cb (y=0) is the depletion capacitance -- F. MOSFET Small Signal Model at Low Frequencygmbgm---------2 q εsNa2 Cox2– φpVBS–----------------------------------------------1Cox---------q εsNa22–φpVBS–()---------------------------------------Cby =0()Cox--------------------== =gatesourcedepletionbulkCb(0)regionchannel_gmvgsrogatesourcedrainsource+_vgsid+_vdsgmbvbsbulkvbs+EECS 6.012 Spring 1998Lecture 10III. MOSFET Small Signal Model at High FrequencyA. Terminal Capacitances• Cgs - Overlap capacitanceCov + Channel charge • Cgd - Overlap capacitanceCov only• Cgb - Only parasitic since bulk charge does not change• Cdb - Drain depletion charge• Csb - Source depletion charge,, ,,,,,,,,,gatedrainsourcen+n+qN(vGS)overlap LD overlap LD fringe electric field linesCsbCdbdepletionregiongmvgsgmbvbsrogatesource__vgsCgsCgbCgdCsbCdbvbsdrainid++bulkEECS 6.012 Spring 1998Lecture 10B. Channel Charge• Recalling that current is related to the electric field• Note bulk Charge is constant with vGS so the channel charge component of Cgs is given by• In saturation the drain has no control over the channel charge so only Cgs has a channel charge component given byqNvGS() W–CoxvGSVTn– vCy()–dy0L∫=dyWCoxµniD----------------------vGSVTn– vC–dvC=qNvGS()W2µnCox2iD--------------------------– vGSVTn– vC–2dvC0vGSVTn–∫=qNvGS()23---– WLCoxvGSVTn–=qGvGS() qNvGS()– qB max,–=dqGdvGS--------------VGS23---WLCox=EECS 6.012 Spring 1998Lecture 10C. Parasitic Capacitance from Source & Drain Depletion Regions• The drain n and p regions have depletion regions whose stored charge changes during the transient.• Depletion qJ(vD) is non-linear --> take the worst case and use the zero-bias capacitance Cjo as a linear charge-storage element during the transient. • Perimeter of the drain diffusion is also important and must be included in the calculation as a capacitance/length x perimeter of diffusion:gate contactWgateinterconnectn+ polysilicon gatesourceinterconnectdraininterconnectsource contactsLdiffArea of drain diffusion:Adiff = W LdiffPerimeter of drain diffusion:P = W + 2 Ldiff(side next to gateisn’t
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