6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 24-1 Lecture 24 - Frequency Response of Amplifiers (II) Open-Circuit Time-Constant Technique December 6, 2005 Contents: 1. Open-circuit time-constant technique 2. Application of OCT to common-source amplifier 3. Frequency response of common-gate amplifier Reading assignment: Howe and Sodini, Ch. 10, §§ 10.4.4-10.4.5. 10.66.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 24-2 Key questions • Is there a fast way to assess the frequency response of an amplifier? • Do all amplifiers suffer from the Miller effect?6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 24-3 1. Open-Circuit Time-Constant Technique Simple technique to estimate bandwidth of an amplifier. Method works well if amplifier transfer function has: • a dominant pole that dominates the bandwidth • no zeroes, or zeroes at frequencies much higher than that of dominant pole Transfer function of form: Vout Avo = Vs (1 + jω ω1)(1 + jω ω2)(1 + jω ω3)... with ω1 � ω1,ω2,ω3, ... log |Av| Avo log ωω1 ω2 -1 -26.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 24-4 Vout Avo = Vs (1 + jω ω1)(1 + jω ω2)(1 + jω ω3)... Multiply out the denominator: Vout Avo = Vs 1+ jωb1 +(jω)2b2 +(jω)3b3... where: 1 1 1 b1 = + + + ... ω1 ω2 ω3 If there is a dominant pole, the low frequency behavior well described by: Vout Avo Avo = Vs �1+ jωb1 1+ jωωH Bandwidth then: 1 ωH �b16.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 24-5 log |Av| Avo log ωω1 ω2 ωH log |Av| Avo log ω -1 -2 -1 It can be shown (see Gray & Meyer, 3rd ed., p. 502) that coefficient b1 can be found exactly through: n nb1 = � τi = � RTiCi i=1 i=1 where: τi is open-circuit time constant for capacitor Ci RTi is Thevenin resistance across Ci (with all other capacitors open-circuited) Bandwidth then: 1 1 1 = = �ni=1 RTiCi ωH �b1 �ni=1 τi6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 24-6 Summary of open-circuit time constant technique: 1. shut-off all independent sources 2. compute Thevenin resistance RTi seen by each Ci with all other C’s open 3. compute open-circuit time constant for Ci as τi = RTiCi 4. conservative estimate of bandwidth: 1 ωH �Στi Vout IoutWorks also with other transfer functions: Iout , Is , Is .Vs6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 24-7 2. Application of OCT to evaluate bandwidth of common source amplifier VDD vs VGG vOUT iSUP RS RL signal source + -signal� load VSS Small-signal equivalent circuit model (assuming current source has no parasitic capacitance): RS Cgd ++ --vgs + -voutgmvgsvs Cgs Rout 'Cdb Three capacitors ⇒ three time constants6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 24-8 � First, short vs: Cgd + -vgs gmvgsCgs Rout 'RS Cdb � Time constant associated with Cgs + -vgs gmvgs Rout 'RS + -vt it Clearly: RTgs = RS and time constant associated with Cgs is: τgs = RSCgs6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 24-9 � Time constant associated with Cdb: + -vgs gmvgs Rout 'RS + -vt it Note: vgs =0 Then: RTdb = R�out and time constant associated with Cgs is: τgs = R�outCdb�6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 24-10 Time constant associated with Cgd: + -vgs gmvgs Rout 'RS + -vt it Note: vgs = itRS Also: vt = vgs +(gmvgs + it)R�out Putting it all together, we have: vt = it[RS + Rout(1 + gmRS)] Then: RTgd = RS + Rout�(1 + gmRS)= Rout� + RS(1 + gmRout�) and time constant associated with Cgd: τgd =[Rout� + RS(1 + gmRout�)]Cgd�6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 24-11 The bandwidth is then: 1 1 =ωH �Στi RSCgs +[R�+ RS(1 + gmR�out out)]Cgd + RoutCdb Identical result as in last lecture. Open circuit time constant technique evaluates bandwidth neglecting −ω2 term in the denominator of Av conservative estimate of ωH.⇒6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 24-12 3. Frequency response of common-gate ampli-fier VDD is iOUT VSS iSUP IBIAS RL signal source signal� load RS VSS Features: • current gain � 1 • low input resistance • high output resistance •⇒good current bufferi6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 24-13 Small-signal equivalent circuit model: G S D + -vgs + -vbs gmvgs ro RLroc Cgd Cgs Csb gmbvbs Cdb is out RS B vgs=vbs (gm+gmb)vgs + -vgs ro Cgs+Csb Cgd+CdbRS roc//RL=RL' is � Frequency analysis: first, open is: (gm+gmb)vgs + -vgs ro C1=Cgs+Csb C2=Cgd+Cdb RL' RS6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 24-14 � Time constant associated with C1: ro (gm+gmb)vgs RL' RS + -vt it (gm+gmb)vgs ro RL' + -vt ' it ' Don’t need to solve: • test probe is in parallel with RS, • test probe looks into input of amplifier ⇒ sees Rin! RT 1 = RS//Rin And: τ1 =(Cgs + Csb)(RS//Rin)6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 24-15 � Time constant associated with C2: ro (gm+gmb)vgs rocRS RL + -vt it (gm+gmb)vgs ro RS + -vt ' it ' roc Again, don’t need to solve: • test probe is in parallel with RL, • test probe looks into output of amplifier ⇒ sees Rout! RT 2 = RL//Rout And: τ2 =(Cgd + Cdb)(RL//Rout)6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 24-16 � Bandwidth: 1 ωH �(Cgs + Csb)(RS//Rin)+(Cgd + Cdb)(RL//Rout) No capacitor in Miller position → no Miller-like term. Simplify: • In a current amplifier, RS � Rin: 1 1 RT 1 = RS//Rin � Rin �gm + gmb �gm • At output: 1 RT 2 = RL//Rout = RL//roc//{ro[1+RS(gm+gmb+ )] ro or RT 2 � RL//roc//[ro(1 + gmRS)] � RL Then: 1 ωH �(Cgs + Csb)g1 m +(Cgd + Cdb)RL If RL is not too high, bandwidth can be rather high (and approach ωT).6.012 - Microelectronic Devices and Circuits - Fall 2005 Lecture 24-17 Key conclusions • Open-circuit time-constant
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