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ECE594I notes, M. Rodwell, copyrighted
ECE594I Notes set 6:Thermal Noise
Mark RodwellUniversity of California, Santa Barbara y
rodwell@ece.ucsb.edu 805-893-3244, 805-893-3262 fax
ECE594I notes, M. Rodwell, copyrighted
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
ECE594I notes, M. Rodwell, copyrighted
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
ECE594I notes, M. Rodwell, copyrighted
Partition Function
sS then, states allowed has system theIf
−= i
i ZkTP )/exp()( εε
∑ −==s
i
kTsEZZ
)/)(exp(function partition where
)(
ECE594I notes, M. Rodwell, copyrighted
Background: Harmonic Oscillator
ff atoscillatorharmonicfrequencyatmodeainPhotons → ff at oscillatorharmonicfrequency at modea in Photons →
ECE594I notes, M. Rodwell, copyrighted
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
ECE594I notes, M. Rodwell, copyrighted
Energy of Photons in Some Mode
)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
ECE594I notes, M. Rodwell, copyrighted
Energy of Photons in Some Mode
⎞⎛ { }( )/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
ECE594I notes, M. Rodwell, copyrighted
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 ⋅⋅=
ECE594I notes, M. Rodwell, copyrighted
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=
ECE594I notes, M. Rodwell, copyrighted
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
ECE594I notes, M. Rodwell, copyrighted
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
ECE594I notes, M. Rodwell, copyrighted
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
ECE594I notes, M. Rodwell, copyrighted
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
ECE594I notes, M. Rodwell, copyrighted
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
ECE594I notes, M. Rodwell, copyrighted
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
ECE594I notes, M. Rodwell, copyrighted
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
ECE594I notes, M. Rodwell, copyrighted
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+
+=
ECE594I notes, M. Rodwell, copyrighted
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
ECE594I notes, M. Rodwell, copyrighted
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
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