Lecture 21 Frequency Response of Amplifiers (I) COMMON-SOURCE AMPLIFIER Outline 1. Intrinsic Frequency Response of MOSFETs 2. Frequency Response of Common- Source Amplifier 3. Miller Effect Reading Assignment: Howe and Sodini, Chapter 10, Sections 10.1-10.4 6.012 Electronic Devices and Circuits-Fall 2000 Lecture 21 1 Summary of Key Concepts • 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 = 2ptT – where tT is the transit time of electrons through the channel • In common-source amplifier, voltage gain rolls off at high frequency because Cgs and Cgd short circuit the input • In common-source amplifier, effect of Cgd on bandwidth is amplified by amplifier voltage gain. • Miller Effect is the effect of capacitance across voltage gain nodes magnified by the voltage gain – trade-off between gain and bandwidth 6.012 Electronic Devices and Circuits-Fall 2000 Lecture 21 2 1. Intrinsic Frequency Response of MOSFET How does one assess the intrinsic frequency response of a transistor? ft º short-circuit current-gain cut-off frequency [GHz] Consider a MOSFET biased in saturation regime with small-signal source applied to gate: vs at input Þ iout at output : transistor effect Þ iin at input : due to gate capacitance iout Frequency dependence : f • Þ iin • Þ ¯ iin iout fT º frequency at which =1 iin 6.012 Electronic Devices and Circuits-Fall 2000 Lecture 21 3 Complete small-signal model in saturation Node 1: iin - jwCgs vgs - jwCgd vgs = 0 Þ iin = jw(Cgs + Cgd )vgs Node 2: iout - gm vgs + jwCgd vgs = 0 Þ iout = (gm - jwCgd )vgs 6.012 Electronic Devices and Circuits-Fall 2000 Lecture 21 4 Current Gain iout gm - jwCgd h21 = = iin jw(Cgs + Cgd ) Magnitude of h21: 2 2 2 gm + w Cgd h21 = w(Cgs + Cgd ) g Low Frequency, w << m Cgd gm h21 » w(Cgs + Cgd ) g High Frequency, w >> m Cgd Cgd h21 » Cgs + Cgd 6.012 Electronic Devices and Circuits-Fall 2000 Lecture 21 5 Current Gain (contd.) |h21| becomes unity at: gm wT = 2pfT = Cgs + Cgd Then: gm fT = 2p(Cgs + Cgd ) 6.012 Electronic Devices and Circuits-Fall 2000 Lecture 21 6 Current Gain (contd…) Phase of h21: g Low Frequency, w << m Cgd p f » - h21 2 g High Frequency, w >> m Cgd f » -p h21 6.012 Electronic Devices and Circuits-Fall 2000 Lecture 21 7 Physical Interpretation of fT: Consider: 1 C + C C = gs gd » gs 2pfT gm gm Plug in device physics expressions for Cgs and gm: 2 LWC 1 C ox L » gs = 3 = 2pf g W 3 V - V T m mC (V - V ) m GS T L ox GS T 2 L or 1 L L » = = tT 2pfT m Echannel vchannel tT º transit time from source to drain [s] Then: 1 fT » 2ptT fT gives an idea of the intrinsic delay of the transistor: Good first order figure of merit fro frequency response 6.012 Electronic Devices and Circuits-Fall 2000 Lecture 21 8 Frequency Response of MOSFET How do we reduce tT and increase fT? • L¯ : trade-off cost • (VGS – VT) • Þ ID• : trade-off power • m• : hard to do • Note: fT is independent of W Impact of bias point on fT: W W mCox (VGS - VT ) 2 mCoxID gm L L fT = = = 2p(Cgs + Cgd ) 2p(Cgs + Cgd ) 2p(Cgs + Cgd ) In a typical MOSFET at typical bias points: fT »1-25 GHz 6.012 Electronic Devices and Circuits-Fall 2000 Lecture 21 9 2. Frequency Response of the Common- Source Amplifier Small-signal equivalent circuit model: Low-frequency voltage gain: vout ' A v,LF = = -gm (ro // roc // RL ) = -gm Rout vs 6.012 Electronic Devices and Circuits-Fall 2000 Lecture 21 10 Node 1: vs - vgs - jwCgs vgs - jwCgd (vgs - vout )= 0 RS vout Node 2: - gm vgs + jwCgd (vgs - vout )- ' = 0 Rout Solve for vgs in 2: 1 jwCgd + R¢out vgs = vout jwCgd - gm Plug vgs in 1 and solve for vout/vs: - (g - jwC )R¢ A = m gd out v ì ü ï é æ 1 öùï 2 1+ jwR S íCgs + Cgd ê1+ R¢out ç + gm ÷úý - w RSR¢outCgsCgd ç R ÷ îï ë è S øûþï Check that for w = 0, Av,LF = -gm R’out 6.012 Electronic Devices and Circuits-Fall 2000 Lecture 21 11 Simplify gm 1. Operate at w << wT = Cgs + Cgd Þ gm >> w(Cgs + Cgd )> wCgs , wCgd 2. Assume gm high enough so that 1 + gm » gm R s 2 3. Compare w term in the denominator of Av with a portion of the w term: w2R R¢ C C wC S out gs gd = gs << 1 wRSCgdR¢outgm gm Then: - gm R¢out A v = 1+ jwRS {Cgs + Cgd [1+ gm R¢out ]} This has the form: A A (w) = v,LF v w 1+ j wH 6.012 Electronic Devices and Circuits-Fall 2000 Lecture 21 12 Amplifier Frequency Response At w = wH: A A (w ) = v,LF v H 2 wH gives an idea of frequency beyond which |Av| starts rolling off quickly Þ bandwidth For the common source amplifier 1 wH = RS [Cgs + Cgd (1+ gm R¢out )] Frequency response of common-source amplifier limited by Cgs and Cgd shorting out the input. 6.012 Electronic Devices and Circuits-Fall 2000 Lecture 21 13 Amplifier Frequency Response (Contd.) We can re-write as: 1 wH = RS [Cgs + Cgd (1+ A v,LF )] To improve bandwidth, • Cgs, Cgd ¯ Þ small transistor with low parasitics • |Av,LF| ¯ Þ do not use more gain than necessary But… Effect of Cgd on wH is being magnified by 1+ | |Av,LF| Why??? 6.012 Electronic Devices and Circuits-Fall 2000 Lecture 21 14 3. Miller Effect In common-source amplifier, Cgd looks much bigger than it really is. Consider a simple voltage-gain stage: What is the input impedance? iin = (vin - vout )jwC But vout = -A v vin Then: iin = jwC(1+ A v )vin 6.012 Electronic Devices and Circuits-Fall 2000 Lecture 21 15 Miller Effect (contd.) or vin 1 Z in = = iin jwC(1+ A v ) Looking in from the input, C appears bigger than it really is. This is called Miller Effect. When a capacitor is located across nodes where there is a voltage gain, its effect on bandwidth is amplified by the voltage gain Þ Miller Capacitance Why? vin • Þ vout = -A v vin •• Þ (vin - vout )•• Þ iin •• In amplifier stages with voltage gain, it is critical to have small capacitance across nodes that have voltage gain. As a result of the Miller effect, there is a fundamental gain-bandwidth trade-off in amplifiers. 6.012 Electronic Devices and Circuits-Fall 2000 Lecture 21 16 What did we learn today? Summary of Key Concepts • fT (short-circuit current-gain cut-off frequency) – frequency at which the short circuit current gain becomes 1. – figure of merit to assess intrinsic frequency response of transistors • In MOSFET, to first order 1 fT = 2ptT – where tT is the transit time of electrons through the channel • In common-source amplifier, voltage gain rolls off at high frequency because Cgs and Cgd short circuit the input • In common-source amplifier, effect of Cgd on bandwidth is amplified by amplifier voltage gain. • Miller Effect is the effect of capacitance across voltage gain nodes magnified by the voltage gain – trade-off between gain and bandwidth 6.012 Electronic Devices and Circuits-Fall 2000 Lecture 21 17.