noise notes set6.ppt...peyton z. peebles : probability, random variables, random signal principles...

ECE594I notes, M. Rodwell, copyrighted

ECE594I Notes set 6:Thermal Noise

Mark RodwellUniversity of California, Santa Barbara y

[email protected] 805-893-3244, 805-893-3262 fax


References and Citations:

DevicesState-Solid in Noise :Zielder Van PhysicsThermal : Kroemerand Kittel

:Citations / Sources

ng.EngineeriionsCommunicatofsPrinciple:Jacobs&Wozencraftory)(introduct s PrincipleSignal RandomVariables, Randomty, Probabili: PeeblesZ. Peyton

ive)comprehens(hard, Variables Randomandity Probabil:Papoulis

1982circaStanfordHellmanMartin:noteslectureyProbabilit1982circa Stanford, Cover, Thomas :notes lecture theory nInformatio

Designc ElectroniNoise Low :erMotchenbakng.EngineeriionsCommunicatofs Principle:Jacobs & Wozencraft

t dffS t d

circuits. in Noise :Notes nsApplicatio tor LinearSemiconduc National1982circa Stanford,Hellman, Martin:notes lecturey Probabilit

design)receiver (optical Personik & Smith noise), (device by Fukui Papers Kroemer and Kittel Peebles,Jacobs, & Wozencraft Ziel,der Van

study.for references Suggested

Theory nInformatio of Elements:Williams andCover )(!Notes App. Semi. National


Boltzmann Law

{ }{ }

{ }{ }

)()(exp/)(exp)()(

/exp )ln(But

201010101 ⎬⎫

⎨⎧ −−−

=−

=−

=

=⇒=

RRRR ESESkESEgP

kSgSgk

εεεεε{ }

)()()()()(But

exp/)(exp)()(

120

20201

−−=−−∂

−=−

⎭⎬

⎩⎨−−

R

RR

ESOESESES

kkESEgP

εεεε

εεε

:enough isorder -1 and small, are esderiviativ thebig, isreservoir theIf

...)( ...)()()(But

st

011010 −−=−−∂

−=−R

RRR TESO

EESES εεε

⎫⎧P )(⎭⎬⎫

⎩⎨⎧ −

=kTP

P 12

2

1 exp)()( εε

εε

not arestatesbecausediffer These Energy.of ondistributiy probabilit not the state, particulara in being )1( freedom of

degree 1 withsystem-suba of probabilty theis this:Important=g

energy. in ddistributeuniformly gy


Partition Function

sS then, states allowed has system theIf

−= i

i ZkTP )/exp()( εε

∑ −==s

i

kTsEZZ

)/)(exp(function partition where

)(


Background: Harmonic Oscillator

ff atoscillatorharmonicfrequencyatmodeainPhotons → ff at oscillatorharmonicfrequency at modea in Photons →


Energy of Photons in Some Mode

.frequency withmode neticElectromag ω

( ) ( )+=+=S shfssE integer. an is where2/12/1:state theof energies Allowed

ωh

{ }∑:function Partition

{ }∑ −=states

s kTEZ /exp



)2/1( ωωω hhh ⎫⎧⎫⎧⎫⎧ + +∞ ss

1

exp2

exp)2/1(exp0

ω

ωωω

h

hhh

⎬⎫

⎨⎧−

⎭⎬⎫

⎩⎨⎧−⋅

⎭⎬⎫

⎩⎨⎧−=

⎭⎬⎫

⎩⎨⎧ +−

= ∑∑= kT

skTkT

sZsstates

{ }/exp11

2exp

ωω

h

h

−−⎭⎬⎫

⎩⎨⎧=

kTkT

exp1

: statea ofoccupancy ofy Probabilitωh⎬⎫

⎨⎧−−

s

2exp

exp1

2expexp)/)(exp()(

ωωω

h

hh

⎭⎬⎫

⎩⎨⎧−

⎭⎬

⎩⎨

⋅⎟⎠⎞

⎜⎝⎛−⋅⎟

⎠⎞

⎜⎝⎛−=

−=

kT

kTkTkT

sZ

kTsEsP

{ }( )/exp1exp

2

ωωh

h−−⋅⎟

⎠⎞

⎜⎝⎛−=

⎭⎩

kTkT

s

kT

. mode in photons )2/1( having of probabilty theis This ωh+⎠⎝

skT



⎞⎛ { }( )/exp1exp)( ωωh

h−−⋅⎟

⎠⎞

⎜⎝⎛−= kT

kTssP

1: of valueExpected

∞

s

1exp

1steps) skip()(][0 ωh −⎟

⎠⎞

⎜⎝⎛

==⋅= ∑=

kT

sPssEs

[ ] for2/1

so ),2/1(But

ωωωω

ω

hhh

h

h

<<→+=+=

+=⎠⎝

kTkTsE

sE

[ ]

ftdthiThi

for 1exp2

2/1 ωω

ω

h

hh

h <<→−⎟

⎠⎞

⎜⎝⎛

+=+= kTkT

kT

sE

.frequency at energy mode averge theis This ωh


Nyquist's Noise Derivation (from Van der Ziel)

T .eTemperatur resistors.matched withline-onTransmissi

VV nn .)(left -right and )(right left-flow to wavea voltagecausesThis voltage.noisea thermal hasresistor Each

21>

fPRjfSjfSf

Pav

/4)(~)(~hiFi l

frequency inresistor each from availablepower Define

ΔΔ

=

fPRjfSjfSf avVVVV nnnn

liP

./4)()(this, From. interval2211

Δ⋅==Δ

vlPP avLine /2line onPower ⋅⋅=


Nyquist's Noise Derivation (from Van der Ziel)

switches. 2 closingby radiation gpropagatin thisTrap

)2/( are sfrequencie allowed closed, switches With lvn

)/2(: bandwidth considered withinsfrequencie allowed #

vlnf#f

Δ=Δ

)/2( vlnf# ⋅Δ=


Nyquist's Noise Derivation ( from Van der Ziel )

⋅=⋅=

)/2( time)on(propagatiresistors) from(Power line inEnergy

vlPAV

( ) ⎥⎤

⎢⎡

+⋅⋅Δ=

⋅Δ=

)/2(

mode)per (energy ) in modes (#line inEnergy

vlf

f

ωω hh

( )

⎤⎡⎤⎡

⎥⎦

⎢⎣ −

+Δ

So1/exp2

)/2( kT

vlfωh

( ) ( ) ⎥⎦

⎤⎢⎣

⎡−

+⋅Δ=⎥⎦

⎤⎢⎣

⎡−

+⋅Δ=

~~1/exp21/exp2

kThf

hfhffkT

fPAV ωωω

h

hh

( ) ⎥⎤

⎢⎡

+⋅=

Δ⋅==

4)(~

/4)(~)(~ since And2211

hfhfRjfS

fPRjfSjfSnnnn

VV

avVVVV

( ) ⎥⎦

⎢⎣ −

+1/exp2

4)(11 kThf

RjfSnn VV


Nyquist's Noise Derivation ( from Van der Ziel )

( ) 1/exp2 ⎥

⎦

⎤⎢⎣

⎡−

+⋅Δ=kThf

hfhffPAV

( ) 1/exp2 ⎥⎦

⎤⎢⎣

⎡−

+=kThf

hfhfdf

dPAV

( ) 1/exp24)(~

343423

11 ⎥⎦

⎤⎢⎣

⎡−

+⋅=kThf

hfhfRjfSnn VV

sJ1006.1 s,J106.6 J/K, 1038.1 343423 ⋅⋅=⋅⋅=⋅= −−− hhk


Comment about Noise Derivation

lththi lih iTh resonator. RLCan uses derivationAnother

difficultyunderlyingsamethehavewecasesbothIn

complex.moremaththesimpler, is physics The

someoverextendsfrequencieallowedandzero,-nonbecomeslinewidthresonator theresistors, toresonator thecoupling In.difficulty underlyingsamethehave wecases both In

.oscillatorharmonicquantuma offrequency single the torestricted being nrather tha bandwidth, smallsomeover extendsfrequencieallowedandzero,non becomes

qq y


Available Thermal Noise Power

2/isloadroughcurrent thload,matchedWith2/ is load across voltageload, matched With

.generator tomatched load :sferpower tran Maximum

N

N

IE

RR

RR

thatGiven

2/isloadroughcurrent th load, matched With N

dk

IEn

4)(~or 4)(~ kTdf

dPRkTjfSkTRjfS load

IIEE nnnn=⇒==

hencepowernoiseavailable)(themaximumtheisP

R In R

hence power,noiseavailable)(the maximum theis

, kTdf

dPP

noiseavailable

load

=

law. thisfollows bias) (no mequilibriu malunder thert compononenAny power.noiseavailableequal have resistors All


Thermal Noise

RR InhfhfRjfS EE *4)(~

⎥⎤

⎢⎡

+=En

kThfhfhf

RjfS

kThfRjfS

nn

nn

II

EE

1)/exp(2*4)(~

1)/exp(24)(

⎥⎦

⎤⎢⎣

⎡+=

⎥⎦

⎢⎣ −

+

kThf

kThfR

become theseFor

1)/exp(2

<<

⎥⎦

⎢⎣ −

kTjfS

kTRjfS

f

nnEE

4)(~

4)(~

=

=Available Power

kTR

jfSnnII )( =

FrequencyhF=kT


Noise from any impedance under thermal equilibrium

supply)energy no ( mequilibriu malunder thernetwork complex or component any For

, kTdf

dP noiseavailable =

)Re(4)(~or )Re(4)(~ YkTjfSZkTjfSdf

nnnn IIEE ==⇒

networks passivecomplex of ncalculatio noisequick allows This amics. thermodynoflaw 2 thefrom follows This nd

m.equilibriuthermalinNOTaredevicestorsemiconducBiased

antennas.of ncalculationoisequick allows This

m.equilibriuthermalinNOTaredevicestor semiconduc Biased


Noise from any impedance: Example

noise calculate tomethodFirst

−+−

+=+

+=22

221

2

21

)1()1)(1(

)1(1

)(

CRjRCRjRCRjCRj

CRjRRCRj

RRjZ

ωωωω

ωω

ω

+=+=

+−

++=

+−

+=

2

222

22

222

22

1222

222

1

)(where)()()(

111)1(

RRjRjjXjRjZ

CRCRj

CRRR

CRCRjRR

ωωωω

ωω

ωωω

=

++=+= 22

221

)(4)(~1

)(where)()()(

jkTRjfS

CRRjRjjXjRjZ

nnVV ω

ωωωωω

⎥⎦

⎤⎢⎣

⎡+

+⋅= 222

22

1 14

CRRRkTω


Noise from any impedance: Example

1noisecalculate tomethod Second

11)()()(

*

22*

221,

ee

CRjjejeje

nn

nntotaln ωωωω

⎟⎟⎞

⎜⎜⎛

⎟⎟⎞

⎜⎜⎛

++=

11*12

*21

222

*22*

11

2

21

2

21,,

eeeeeeee

CRje

CRjeee

nnnnnnnn

nn

nntotalntotaln ωω

+++=

⎟⎟⎠

⎜⎜⎝ +

+⋅⎟⎟⎠

⎜⎜⎝ +

+=

)(~tindependen are processes thebecause 0But

111*

21*

21

2222

2211

jfS

eeeeCRjCRjCR

ee

nnnn

nn ωωω

==

+−+

( )44

21)(

)(~)(~

2

222

22,2,

1,1,,,

kTRkTR

CRfjfS

jfSjfS nn

nntotalntotaln

EEEEEE π+

+=

( )answer. same....

214 22

22

21 CRf

kTRπ+

+=


Noise from an Antenna

)Re(4)(~ ,EE

noiseavailable ZkTjfSkTdf

dPnn

=⇒=

s.resistance radiation and Ohmic both hasantenna The

Rrad

mperatureantenna te physical theis where, 4density spectralofvoltagenoisea has resistance Ohmic The

ambientOhmicambient TRkT En,rad

Rloss

i lihhi hfihfre temperatuaverage theis where, 4density

spectral of voltagenoisea has resistance radiation thelaw,2 By the

nd

fieldradfield TRkTEn,loss

. Kelvin...3.8at is space galactic-Inter

power signalreceivesantenna the whichfrom region theof


Noise on a capacitor

4)(~or4)(~ From

==kTjfSkTRjfS

RC Vc

+

~11~

that find We

)(or 4)(

*

⎟⎞

⎜⎛⎟⎞

⎜⎛

RjfSkTRjfS

nnnn IIEE

En

C

-

)(~1

)(211

211)(

2222 ⎟⎟⎞

⎜⎜⎛

=

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛+

=

jfS

jfSfRCjfRCj

jfSnncc

EE

EEVV ππ

[ ] ~1

isenergy Capacitor stored mean theSo

)(41 2222

∫∞

⎟⎞

⎜⎛

⎟⎟⎠

⎜⎜⎝ +

jfSCRf nnEEπ

[ ] 2/)(41

1)2/1(0 2222∫ =⎟⎟

⎠

⎞⎜⎜⎝

⎛+

=⋅ kTdfjfSCRf

VVECnnEEcc π

law.Boltzmannthefromdirectly follows also This

noise notes set6.ppt...peyton z. peebles : probability, random variables, random signal principles...

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