Noise02s Raf

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Rafael J. Betancourt-Zamorahttp://www.betasoft.org/

May 16, 2002

Noise Tutorial:

Low-frequency CMOSAnalog Design


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Noise Tutorial: Low-frequency CMOS Analog Design

2002, Rafael J. Betancourt-Zamora http://www.betasoft.org/

Outline

Goals

Noise Overview

Resistor Noise Sources

MOSFET Noise Sources

Noise of Common Amplifier Topologies

Examples

Summary


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Grasp the importance of noise performance in low-frequency

instrumentation design

Understand the types of noise present in integrated resistorsand MOS devices.

Understand the noise trade-offs of different circuit topologies

Perform simple noise analysis

Goals


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Outline

Goals

Noise Overview




Examples

Summary


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t

V t( )

P x( )

FUZZ 6

Random process with zero mean

Instantateous amplitude is unpredictable

Average noise power can be measured (variance)

What does it look like?

WHITE NOISE (GAUSSIAN DISTRIBUTION)

m 0=

Vrms

=

STANDARD DEVIATION

VARIANCE

MEAN

2

Vrms

2

=f

Vrm s2

Hz

Noise Power Spectral Density(Normalized Noise Power)


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Types of Noise

THERMAL

SHOT

FLICKER

thermal excitation of charge carriers

fluctuations in dc current flow through junctions

traps in semiconductors

f

Vrm s2

Hz

f

Vrm s2

Hz

f

Arms

2Hz

1 f

Vrms2

f---------- 4kTR=

Irms2

f----------- 2qIDC=

Vrms2

f-------------

Kf VDC2

f--------------------=

, V2/Hz

, A2/Hz

, V2/Hz

k= 1.38 x 10-23J/K

q= 1.6 x 10-19C

T= Temperature, K

Kf= Flicker coefficient


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Resistor Thermal Noise

Vn2

f------ 4kTR= , V2/Hz

R

* R

Vn2 4kTRfn=

Vn2

In2 4kT

R

---------------- fn=

In2

, V2 , A2

NOISE DENSITY

NOISE POWER

In2

f-------

4kTR

----------------= , A2/Hz

1k@ 25 C

nV Hz

pA Hz

4.05

4.05

fn = NOISE BANDWIDTH

Integrate noise PSD over frequency to arrive at total noise power.

For white noise, multiply PSD by noise bandwidth.


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Noise Bandwidth

NOISE BW

Noise bandwidth is defined for a brickwall transfer function

Noise bandwidth is not the same as 3dB bandwidth

ff3dB fn

Avo2

Avo2

fn Avf( )2

0

df=

No. Poles

1

2

3

2nd-order BPF

fn2--- f3dB=

fn 1.22 f3dB=

fn 1.16 f3dB=

fn2--- f3dB=

Avf( )

2


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kT/C Limit

C

R

*

Vn2

Vo2

fn2--- f3dB=

f3dB1

2RC---------------=

fn1

4RC-----------=

Vo2 4kTRfn= , V2

}

Vo2 4kTR 1

4RC-----------

=

Vo2 kT

C------=

Total noise power is independent of R

Capacitor is noiseless, but accumulates noise from resistor

This fundamental limit is important for sampled systems

, V2


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Combining Noise Sources

R1

*R2Vn1

2 In22

R2

*Vn2

2

R1+R2

* Vn1

2Vn2

2+

R1 In12

R1||R2In1

2In2

2+

For uncorrelated noise sources, just add the noise powers.

For correlated sources use:

Vo2

Vn12

Vn22

2cVn1Vn2+ +=

c 1 CORRELATION COEFFICIENT

SERIES PARALLEL

Vo2 4kT R1 R2+( )fn= Io

24kT 1

R1-------

1

R2-------+( )fn=


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Resistor Flicker (1/f) Noise

f

Vn2

Hz1 f Vnf

2

f-------

Kf

VDC2

f---------------------= , V

2/HzTHERMAL

fc

fff Vnf2 Kf

VDC2

f---------------------

fL

fH

df K

f VDC

2 fH

fL-----

ln= =

Vo2 4kTRfn K

f VDC

2 fHfL------

ln+=

TOTAL RESISTOR NOISE

, V2

Constant noise power per decade

fc

defines 1/fcorner frequency where

flicker noise is equal to thermal noise.

Vnf2

2.3Kf VDC2= FORfH/fL= 10

fL fH

, V2

ASSUMES

fn fc


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Noise Index

Vnf2

VDC2

-------------

DEC

Kf VDC2 10( )ln

VDC2

----------------------------------------------------- 2.3Kf= =

Normalize 1/fnoise per Volt, per decade of frequency

Can express 1/fnoise PSD and fcin terms of NI

NI 2.3K

f

10 6

= , V/V/decade

Vnf2 NI

2VDC

2

2.3 10 12

-----------------------------

1

f---= , V2/Hz

fc

NI2

VDC2

2.3 10 12

4kTR--------------------------------------------------=

, Hz20log(NI)

poly

pdiff

ndiff -10dB

-5dB

-15dB

[Motchenbacher93]


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Outline

Goals

Noise Overview


MOSFET Noise Sou rces


Examples

Summary


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Noise Sources in a MOSFET

*

Vnf2 Ind2

*

Vng2DRAIN THERMAL NOISE

FLICKER NOISE

GATE THERMAL NOISE

Vng2

f-------- 4kTRG 3=

Ind2

f------- 4kTgm=

, V2/Hz

, A2/Hz

={{2/3, long channel

2-3, short channel

Rds = 1/gm

Long Channel

Saturation

due to limited channel conductance

can be neglected with good layout

Vnf2

f-------

Kf

WLCoxf-------------------= , V

2/Hz Kf 1.2 10

24= V2 FFOR NMOS

[Razavi01]

SPICE

Inf2

f------

Kf

IDCAF

LEFF2

Coxf-----------------------= , A

2/Hz


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Input-referred Noise

*

Vnf2 Ind2

*

Vni2

Ini2

Define equivalent input voltage andcurrent noise sources.

MOSFET input current noise sourcecan be neglected at low frequencies.

*

Vni2

Vnd2

f--------

Ind2

f------- gm

2 4kTgm

---------------------= = , V2/Hz

INPUT-REFERRED DRAIN NOISE

TOTAL INPUT-REFERRED NOISE

Vni2

f-------

4kTgm

---------------------

Kf

WLCoxf-------------------+= , V

2/Hz


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1/f Corner Frequency

4kTgm

---------------------

Kf

WLCoxfc----------------------=

f

Vn2 Hz

1 f

THERMAL

fc

fff

Defines where flicker noise is equal to thermal noise.

Need to increase transistor area to reduce flicker noise.

fc

Kfgm4kTWLCox---------------------------------------=


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Outline

Goals

Noise Overview



Noise of Common Ampl i fier Topolog ies

Examples

Summary


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Current Source

In1

2

I1

VBIAS

In12

f------- 4kTgm=

Neglected flicker noise of M1

Need to minimize output current noise

Minimize gmfor low noise

Vo2

Vo2

f

--------In1

2

f

------- ro2

= , V2/Hz

, A2/Hz

VDD

M1

Vo2

f-------- 4kTgmro

2= , V

2/Hz


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Common Source

RL

VDD

M1

InL2

*

Vnf2 In12

Vo2

Vi

Neglected output conductance of M1

Maximize gm, RL, and transistor area for low noise

For a differential pair, double the noise

Vni2

f-------

4kTgm

---------------------

Kf

WLCoxf-------------------

4kT

gm2RL

----------------------+ += , V2/Hz

In12

f------- 4kTgm= , A

2/Hz

InL2

f---------

4kTRL----------------= , A

2/Hz Vnf

2

f-------

Kf

WLCoxf-------------------= , V

2/Hz


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Source Follower

M1

In22

Vo2

*

Vn12

VDD

Vi

VBIAS

M2

Neglected flicker noise of M2

VERY NOISY, AVOID

Vni2

f-------

Vn12

f--------

In22

f-------

1gm1--------------- ro1

2ro2

2|| ||

2

Av

2---------------------------------------------------------------------------+= , V2/Hz

Vn12

f--------

4kTgm1---------------------

Kf

WLCoxf-------------------+= , V

2/Hz

In22

f------- 4kTgm= , A

2/Hz

Av

1WHERE


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Common Gate

In12

Vo2

VBIAS

M1

RL

VDD

InL2

Vi


Define equivalent input voltage and current noise sources

Input noise current cannot be ignored

Transfers load current noise to the input

InL2

f--------- 4kT

RL----------------= , A

2/Hz

In12

f------- 4kTgm= , A

2/Hz

Ini2

f-------

4kTR

L

----------------gm

2Kf

WLCoxf-------------------+= , A

2/Hz

Vni2

f-------

4kTgm

---------------------4kT

gm2RL

----------------------

Kf

WLCoxf-------------------+ += , V

2/Hz

Rs = 1/gm

(neglecting

body effect)

*

Vnf2

Vnf2

f-------

Kf

WLCoxf-------------------= , V

2/Hz


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Cascode

Vo2

VBIAS

M1

RL

VDD

InL2

*

Vn12

Vi

In22M2

Vn12

f--------

4kTgm1---------------------

Kf

WLCoxf-------------------+= , V

2/Hz

InL2

f---------

4kT

RL

----------------= , A2/Hz


M2s noise contribution is negligible at low frequencies

Capacitance at M1s drain increase noise at high frequencies

Vni2

f-------

4kTgm1---------------------

Kf

WLCoxf-------------------

4kT

gm12

RL

--------------------------+ += , V2/Hz


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Outline

Goals

Noise Overview


MOSFET Noise Sources Noise of Common Amplifier Topologies

Examples

Summary


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Variable Resistor

(1-k)RTIn2

2

kRT In12

Vo

Vi

(1-k)RT

In2

kRT

Vo

Req= k(1-k)RTWORST CASE

k= 0.5Req=RT/4

In2

f

------- 4kT

Req----------------= , A

2/Hz

In2

f-------

16kTRT

---------------------= , A2/Hz


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Inverting Operational Amplifier Filter

R1 * R2 *

C

*

Vn12

Vn22

Vo

Vi

Vna2

-

+

Negligible input noise current

Ignore 1/f noise

Low frequency operation

where

Vni2

f------- Vn1

2 Vn22

Avi2

--------+=

Avi

R2

R1-------=

1. SHORT Vna

2. SHORT Vi

Vni2

f------- Vna

2 A

vni2

Avi2

------------

Vna2

1 R1

R2-------+

2

= =

Avni

1 R2

R1-------+=

3. APPLY SUPERPOSITION

Vni2

f------- Vn1

2 Vn22

Avi2

--------+ Vna2

1 R1

R2-------+

2

+=

FOR LARGE GAIN, R2 >> R1

Vni2

f------- Vn1

2 Vn22

Avi2

--------+ Vna2

+= , V2/Hz

fn2---

1

2R2C----------------- 1

4R2C-------------= =


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Outline

Goals

Noise Overview


MOSFET Noise Sources Noise of Common Amplifier Topologies

Examples

Summary


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What We Learned

Thermal noise and flicker noise are the major sources of headachesfor low-noise, low frequency instrumentation design in CMOS.

Noise bandwidth is not the same as 3dB bandwidth.

Capacitors are noiseless, but they can accumulate noise.

Resistors can have flicker noise, but only when there is current flow.

NOISE IN GENERAL

NOISE IN MOSFETS

Need to increase transistor area to reduce flicker noise.

Current source: Minimize gmfor low noise current

Common source: Maximize gm, RL, and transistor area for low noise

Source follower: VERY NOISY, AVOID Common gate: Transfers load current noise to the input. Input noise

current cannot be ignored.

Cascode: Cascode transistor noise contribution is negligible at lowfrequencies.

N i T i l L f CMOS A l D i


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1. P.H. Garret, Analog I/O Design: Acquisition, Conversion,Recovery,Reston Publishing, Reston, VA, pp. 264, 1981.

2. D.A. Johns, K. Martin Analog Integrated Circuit Design,John Wiley & Sons, pp. 706, 1997.

3. C.D. Motchenbacher, J.A. Connelly, Low Noise ElectronicSystem Design,John Wiley & Sons, pp. 422, 1993.

4. B. Razavi, Design of Analog CMOS Integrated Circuits,McGraw-Hill, pp. 684, 2001.

References

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