University:
Massachusetts Institute of Technology
Course:
6.012 | Microelectronic Devices and Circuits
Academic year:

2019
Views:

57

Pages:

18
Author:

Sporkz

Lecture 23 - Frequency Response of Ampliﬁers (I) Common-Source Amplifier December 1, 2005 Contents: 1. Introduction 2. Intrinsic frequency response of MOSFET 3. Frequency response of common-source ampliﬁer 4. Miller eﬀect Reading assignment: Howe and Sodini, Ch. 10, §§10.1-10.4 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 analog 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 LMDS Video Spacewav WirelessMAN Skybridge 4 MMDS 3G 0 0 2 8 Teledesic 20 40 45 BW (MHz) Figure by MIT OCW. 100 155 500 2. Intrinsic frequency response of MOSFET 2 How does one assess the intrinsic frequency response of a transistor? short-circuit current-gain -g cut-oﬀ frequency [GHz] ft ≡ short-circuit Consider MOSFET biased in saturation regime with smallsignal 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 small-signal 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 − gmvgs + vgsjωCgd = 0 ⇒ iout = vgs(gm − jωCgd ) Current gain: iout gm − jωCgd h21 = = iin jω(Cgs + Cgd ) 2 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 <1 |h21| 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 2 Physical interpretation of fT : Consider: Cgs + Cgd Cgs 1 = 2πfT gm gm Plug in device physics expressions for Cgs and gm: 1 Cgs = 2πfT gm 2 LW Cox 3 W µCox(VGS L − VT ) = L µ 32 VGSL−VT or L L 1 = = 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 ﬁrst-order ﬁgure of merit for frequency response. To reduce τt and increase fT : • L ↓: trade-oﬀ is cost • (VGS − VT ) ↑⇒ ID ↑: trade-oﬀ 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 GH z ID ft of different device technologies 600 Cutoff Frequency [GHz] 500 400 c d e f c d Fujitsu (02, EDL) UIUC (03, EL) IBM (03, IPRM) IBM (04, VLSI) e III-V HEMT 300 III-V HBT f 200 SiGe HBT 100 0 1985 Si CMOS 1990 1995 Year 2000 2005 3. Frequency response of common-source amp VDD iSUP signal source RS signal load + vOUT vs RL - VGG VSS Small-signal equivalent circuit model (assuming current source has no parasitic capacitance): Cgd RS + + vgs vs - + Cgs gmvgs Cdb ro roc RL - - Rout' Low-frequency 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 −voutjωCdb − Rvout = 0 out Solve for vgs in 2: vgs = vout jω(Cgd + Cdb ) + R1 jωCgd − gm out Plug in 1 and solve for vout/vs: Av = −(gm − jωCgd )Rout DEN with DEN = 1 + jω{RS Cgs + RS Cgd [1 + Rout ( −ω 2RS RoutCgs (Cgd + Cdb ) 1 + gm)] + Rout Cdb} RS [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 ⇒ worst-case estimation of bandwidth Then: −gmRout Av 1 + jω[RS Cgs + RS Cgd (1 + gmRout) + RoutCdb ] 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 common-source ampliﬁer: ωH = 1 ) + R C RS Cgs + RS Cgd (1 + gmRout out db Frequency response of common-source ampliﬁer limited by Cgs and Cgd shorting out the input, and Cdb shorting out the output. Can rewrite as: 1 fH = 2π{RS [Cgs + Cgd (1 + |Av,LF |)] + RoutCdb } Compare with: fT = gm 2π(Cgs + Cgd ) 2 In general: fH fT due to • typically: gm 1 RS • Cdb enters fH but not fT • presence of |Av,LF | in denominator 2 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 ampliﬁed by 1 + |Av,LF |??!! 4. Miller eﬀect In common-source ampliﬁer, Cgd looks much bigger than it really is. Consider simple voltage-gain 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 = vin 1 = 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 Mil ler ccappacitance: acitance: 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. fundamental As a result of the Miller eﬀect, there is a fundamental gain-bandwidth gain-bandwidth tradeoﬀ in ampliﬁers. Key conclusions • fT (short-circuit current-gain cut-oﬀ frequency): ﬁgure 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 common-source ampliﬁer, voltage gain rolls oﬀ at high frequency because Cgs and Cgd short out input and Cdb shorts out output. • In common-source ampliﬁer, eﬀect of Cgd on bandwidth is magniﬁed by ampliﬁer voltage gain. • Miller eﬀect: eﬀect of capacitance across voltage gain nodes is magniﬁed by voltage gain ⇒ trade-oﬀ between gain and bandwidth.