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Lecture 21 Frequency Response of (I) COMMON-SOURCE

Outline

1. Intrinsic Frequency Response of 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 • 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 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 ?

ft º short-circuit current-gain cut-off frequency [GHz]

Consider a MOSFET biased in saturation regime with small- 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 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