Lecture 23 ­- Frequency Response of Amplifiers (I): Common-­Source Amplifier

Lecture 23 ­ Frequency Response of Amplifiers (I) Common­Source Amplifier December 1, 2005 Contents: 1. Introduction 2. Intrinsic frequency response of MOSFET 3. Frequency response of common­source amplifier 4. Miller effect Reading assignment: Howe and Sodini, Ch. 10, §§10.1­10.4 Key questions • How does one assess the intrinsic frequency response of a transistor? • What limits the frequency response of an amplifier? • What is the ”Miller effect”? 1. Introduction Frequency domain is a major consideration in most ana­ log circuits. Data rates, bandwidths, carrier frequencies all pushing up. Motivation: • Processor speeds ↑ • Traffic volume ↑ ⇒ data rates ↑ • More bandwidth available at higher frequencies in the spectrum DOM Radio 'V Band' 60 Frequency 50 40 25 20 WE Datacom Video Teledesic Spacewav WirelessMAN Skybridge 4 MMDS 3G 0 0 2 8 LMDS 20 40 45 100 BW (MHz) Figure by MIT OCW. 155 500 2. Intrinsic frequency response of MOSFET � How does one assess the intrinsic frequency response of a transistor? ft ≡ short­circuit current­gain cut­off frequency [GHz] Consider MOSFET biased in saturation regime with small­ signal source applied to gate: VDD iD=ID+iout iG=iin vs VGG vs at input ⇒ iout : transistor effect ⇒ iin due to gate capacitance Frequency dependence: f ↑⇒ iin ↑⇒ | iiout |↓ in iout ft ≡ frequency at which | | = 1 iin Complete small­signal model in saturation: iin + + D vgs vs - iout Cgd G Cgs gmbvbs gmvgs ro - Cdb S vbs Csb + B vbs=0 iin + vs 1 Cgd iout 2 + vgs - Cgs gmvgs - node 1: iin − vgsjωCgs − vgsjωCgd = 0 ⇒ iin = vgsjω(Cgs + Cgd ) node 2: iout − gm vgs + vgsjωCgd = 0 ⇒ iout = vgs(gm − jωCgd ) Current gain: gm − jωCgd iout h21 = = iin jω(Cgs + Cgd ) � Magnitude of h21: � 2 + ω2C 2 gm gd |h21| = ω(Cgs + Cgd ) • For low frequency, ω � |h21| � gm , Cgd gm ω(Cgs + Cgd ) • For high frequency, ω � gm Cgd , Cgd |h21| � <1 Cgs + Cgd log |h21| -1 1 ωT Cgd Cgs+Cgd |h21| becomes unity at: gm ωT = 2πfT = Cgs + Cgd Then: gm fT = 2π(Cgs + Cgd ) log ω � Physical interpretation of fT : Consider: Cgs + Cgd Cgs 1 = � 2πfT gm gm Plug in device physics expressions for Cgs and gm: 1 Cgs � = gm 2πfT 2 LW Cox 3 W µCox(VGS L − VT ) = L µ 32 VGSL−VT or 1 L L = = τt � 2πfT µ < Echan > < vchan > τt ≡ transit time from source to drain [s] Then: fT � 1 2πτt fT gives an idea of the intrinsic delay of the transistor: good first­order figure of merit for frequency response. To reduce τt and increase fT : • L ↓: trade­off is cost • (VGS − VT ) ↑⇒ ID ↑: trade­off is power • µ ↑: hard to do • note: fT independent of W Impact of bias point on fT : gm fT = = 2π(Cgs + Cgd ) 2 WL µCox ID − VT ) = 2π(Cgs + Cgd) 2π(Cgs + Cgd ) W L µCox (VGS fT fT 0 0 VT � VGS 0 In typical MOSFET at typical bias points: fT ∼ 5 − 50 GHz ID 3. Frequency response of common­source amp VDD iSUP signal source RS signal� load + vOUT vs RL - VGG VSS Small­signal equivalent circuit model (assuming current source has no parasitic capacitance): Cgd RS + + vgs vs - + Cgs gmvgs Cdb ro roc RL - - Rout' Low­frequency voltage gain: Av,LF = vout vout � = −gm(ro //roc //RL) = −gmRout vs RS + 1 - node 1: node 2: 2 + + vgs vs Cgd Cgs gmvgs Cdb Rout' vout - vs −vgs RS - − vgsjωCgs − (vgs − vout)jωCgd = 0 (vgs −vout )jωCgd −gmvgs −vout jωCdb − Rvout =0 � out Solve for vgs in 2: vgs = vout jω(Cgd + Cdb ) + R1� out jωCgd − gm Plug in 1 and solve for vout /vs: � −(gm − jωCgd )Rout Av = DEN with � ( DEN = 1 + jω{RS Cgs + RS Cgd [1 + Rout 1 � Cdb} + gm)] + Rout RS � −ω 2RS Rout Cgs (Cgd + Cdb ) � [check that for ω = 0, Av,LF = −gmRout ] Simplify: 1. Operate at ω � ωT = gm Cgs +Cgd ⇒ gm � ω(Cgs + Cgd ) > ωCgs , ωCgd 2. Assume gm high enough so that 1 + gm � gm RS 3. Eliminate ω 2 term in denominator of Av ⇒ worst­case estimation of bandwidth Then: � −gmRout Av � � ) + R� C ] 1 + jω[RS Cgs + RS Cgd (1 + gm Rout out db This has the form: Av (ω) = Av,LF 1 + j ωωH log |Av| gmRout' -1 ωH log ω At ω = ωH : 1 |Av (ωH )| = √ |Av,LF | 2 ωH gives idea of frequency beyond which |Av | starts rolling off quickly ⇒ bandwidth For common­source amplifier: ωH = 1 � ) + R� C RS Cgs + RS Cgd (1 + gm Rout out db Frequency response of common­source amplifier limited by Cgs and Cgd shorting out the input, and Cdb shorting out the output. Can rewrite as: 1 fH = � C } 2π{RS [Cgs + Cgd (1 + |Av,LF |)] + Rout db Compare with: fT = gm 2π(Cgs + Cgd ) � In general: fH � fT due to • typically: gm � 1 RS • Cdb enters fH but not fT • presence of |Av,LF | in denominator � To improve bandwidth, • Cgs, Cgd , Cdb ↓ ⇒ small transistor with low parasitics • |Av,LF | ↓⇒ don’t want more gain than really needed but... why is it that effect of Cgd on fH appears to being am­ plified by 1 + |Av,LF |??!! 4. Miller effect In common­source amplifier, Cgd looks much bigger than it really is. Consider simple voltage­gain stage: iin C + + - vin + - Avvin vout - What is the input impedance? iin = (vin − vout)jωC But vout = −Av vin Then: iin = vin(1 + Av )C or Zin = 1 vin = iin jω(1 + Av )C From input, C, looks much bigger than it really is. This is called the Miller effect. When a capacitor is located across nodes where there is voltage gain, its effect on bandwidth is amplified by the voltage gain ⇒ Miller capacitance: CM iller = C(1 + Av ) Why? vin ↑ ⇒ vout = −Av vin ↓↓ ⇒ (vin − vout) ↑↑ ⇒ iin ↑↑ In amplifier stages with voltage gain, it is critical to have small capacitance across voltage gain nodes. As a result of the Miller effect, there is a fundamental gain­bandwidth tradeoff in amplifiers. Key conclusions • fT (short­circuit current­gain cut­off frequency): fig­ ure of merit to assess intrinsic frequency response of transistors. • In MOSFET, to first order, 1 ft = 2πτt where τt is transit time of electrons through channel. • In common­source amplifier, voltage gain rolls off at high frequency because Cgs and Cgd short out input and Cdb shorts out output. • In common­source amplifier, effect of Cgd on band­ width is magnified by amplifier voltage gain. • Miller effect: effect of capacitance across voltage gain nodes is magnified by voltage gain ⇒ trade­off between gain and bandwidth.