Lecture 23 - Frequency Response of Amplifiers (I) Common-Source Amplifier. December 1, 2005

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:

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 MMDS 3G Skybridge LMDS Spacewav WE Datacom Frequency 0 0 4 20 25 40 50 60 2 8 20 40 45 100 155 500 Video WirelessMAN Teledesic 'V Band' DOM Radio BW (MHz)

cuit current-gain cut-off frequency

2. Intrinsic frequency response of MOSFET

2 How does one assess the intrinsic frequency response of a transistor?

f_t ≡ short-cirshort-circuit current-gain cut-off frequency-g [GHz] Consider MOSFET biased in saturation regime with small-signal source applied to gate:

V_DD

i_D=I_D+i_out i_G=i_in

v_s V_GG

v_s at input ⇒ i_out: transistor effect

⇒ iin due to gate capacitance

|i iout_in | ↓ Frequency dependence: f ↑⇒ iin ↑⇒ i_out f_t ≡ frequency at which | | = 1 i_in

Complete small-signal model in saturation: G S D + -v_{gs Cgs} C_gd C_db C_sb g_mv_gs g_mbv_bs ro + v_bs -iout v_s + -iin B v_bs=0 + + -v_{gs gmvgs} i_out iin v_{s Cgs} C_gd 1 2 node 1: i_in − v_gsjωC_gs − v_gsjωC_gd = 0 ⇒ iin = vgsjω(Cgs + Cgd) node 2: i_out − g_mv_gs + vgsjωCgd = 0 ⇒ iout = vgs(gm − jωCgd)

� Current gain: i_out g_m − jωC_gd h₂₁ = = i_in jω(C_gs + Cgd) 2 Magnitude of h21: g2 + ω2C2 |h21| = _ω(Cm gd gs + Cgd)

• For low frequency, ω  gm _,

C_gd

g_m |h21| 

ω(C_gs + C_gd) • For high frequency, ω  g_m

,

C_gd

|h21|  _C Cgd < 1

log |h₂₁| log ω ωT 1 Cgd C_gs+C_gd -1

|h21| becomes unity at:

g_m ω_T = 2πf_T = C_gs + C_gd Then: g_m f_T = 2π(C_gs + C_gd)

2 Physical interpretation of fT: Consider: 1 2πf_T = C_gs + C_gd g_m
C_gs g_m

Plug in device physics expressions for C_gs and g_m: 1 2πf_T
C_gs g_m = 2 3LW Cox W L µCox(VGS − VT) = L µ3₂VGS−VT L or 1 2πf_T
L µ < E_chan > = L < v_chan > = τt

τ_t ≡ transit time from source to drain [s] Then:

f_T
1 2πτ_t

f_T gives an idea of the intrinsic delay of the transistor: good first-order figure of merit for frequency response.

transit time

�

To reduce τ_t and increase f_T : • 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 :

g_m W _L µC_oxI_D f_T = = L µCox(VGS − VT ) = 2 W 2π(C_gs + C_gd) 2π(C_gs + C_gd) 2π(C_gs + C_gd) f_{T fT} 0 0 V_{T VGS} 0 _I D

In typical MOSFET at typical bias points: f_T ∼ 5 − 50 GH z

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3. Frequency response of common-source amp V_DD vs VGG vOUT i_SUP R_S R_L signal source + -signal load V_SS

Small-signal equivalent circuit model (assuming current source has no parasitic capacitance):

+ + -v_gs + -vout gmvgs v_{s Cgs} Cgd RL roc r_o RS Cdb ' R_out

Low-frequency voltage gain:

A_v,LF = vout = −gm(ro//roc//RL) = −gmR _out

C_gd R_S 1 2 + + -v_gs + -v_out g_mv_gs v_{s Cgs Cdb Rout '} node 1: vs_R−vgs − vgsjωCgs − (vgs − vout)jωCgd = 0 S

node 2: (v_gs−v_out)jωC_gd−g_mv_gs−v_outjωC_db− vout _{= 0}

R _out Solve for v_gs in 2: 1 jω(C_gd + Cdb) + _R v_gs = v_out jωC_gd − g_m out

Plug in 1 and solve for vout/vs:

−(gm − jωCgd)R A_v = out DEN with DEN = 1 + jω{RSCgs + RSCgd[1 + R 1 + gm)] + R out( outCdb} RS −ω2RSR_out Cgs(Cgd + Cdb)

Simplify:

gm

1. Operate at ω  ω_T = _C

gs+C_gd ⇒

g_m  ω(C_gs + C_gd) > ωC_gs, ωC_gd 2. Assume g_m high enough so that

1

+ gm  gm

R_S

3. Eliminate ω2 term in denominator of Av

⇒ worst-case estimation of bandwidth Then:

R −gm

A_v  out

1 + jω[RSCgs + RSCgd(1 + gmR_out ) + R_out Cdb]

This has the form:

A_v(ω) = A

1 + j _ωω

H

log |A_v| g_mR_out' logω ω_H -1 At ω = ω_H: |Av(ωH)| = 1 √ 2|Av,LF|

ω_H gives idea of frequency beyond which |A_v| starts rolling off quickly ⇒ bandwidth

For common-source amplifier:

ω_H = 1

R_SC_gs + R_SC_gd(1 + g_mR_out ) + R_outC_db

Frequency response of common-source amplifier limited by C_gs and C_gd shorting out the input, and C_db shorting out the output.

Can rewrite as: 1 f_H = 2π{R_S [C_gs + C_gd(1 + |A_v,LF |)] + R _outC_db} Compare with: g_m f_T = 2π(C_gs + Cgd) 2 In general: fH  fT due to • typically: gm  _R1 S

• 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

4. Miller effect

In common-source amplifier, Cgd looks much bigger than

it really is.

Consider simple voltage-gain stage:

i_{in C} + + -v_in + -v_out A_vv_in

What is the input impedance?

i_in = (v_in − v_out)jωC But

v_out = −Avvin

Then:

ler ca acitance fundamen andwidth tradeoff or v_in 1 Z_in = = i_in 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 ⇒ Mil Miller c pacitance: p :

C_{M iller} = C(1 + Av)

Why?

v_in ↑ ⇒ v_out = −Avvin ↓↓ ⇒ (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 tal gain-b in amplifiers.

fundamental

Key conclusions

• fT (short-circuit current-gain cut-off frequency):

fig-ure of merit to assess intrinsic frequency response of transistors.

• In MOSFET, to first order, 1 f_t =

2πτ_t

where τ_t is transit time of electrons through channel. • In common-source amplifier, voltage gain rolls off at high frequency because C_gs and C_gd short out input and C_db shorts out output.

• In common-source 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