noise notes set9.ppt - web.ece.ucsb.edu

ECE594I notes, M. Rodwell, copyrighted

ECE594I Notes set 9:Bipolar Transistor Noisep

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


Bipolar Transistor AC Characteristics

)1()0()0(' thenare ionsconcentratcarrier minority base The

./ and / Assume

// ≅−=−=

>>>>

kTqVpo

kTqVpopo

cbbe

enennnn

qkTVqkTV

bebe

0)1()('

)()()(/ ≅−≅−= −

pokTqV

pob

popopo

nenWn cb



0' '

/ 2/ 2/2/ :modeldiffusioncarrier -minority Apply the

22/

11 ≅=≅=

===kTqV

po

nbbb

nVennV

qADWRqAWCqAWGbe

'2~ 2'2~ 11

11 ⎟⎟⎠

⎞⎜⎜⎝

⎛⋅=⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅=

b

nI

R

bI

p

WADqnqSAWqnqS

GRGR τ2 ⎠⎝⎠⎝ bR Wτ



regions. depletioncollector -base and base-emitter thecrossing currents *particle* electron theare ~ and ~ that Note EC II

currents.collector andemitter total the, and fromdistinct are These EC II

time. transit and ecapacitanc both from arise currentsnt displaceme thewhereregions, depletion thein currentsnt displaceme from arises distintion The


BJT Characteristics: Space Charge Transit Times

withcapacitoraofplatesbetweenyat velocitmovingelectronanWith v

∫ by the done work the to)( field electric theit through movingby

electron theon done work theequating , voltagebias and separationwithcapacitor a ofplatesbetweeny at velocit moving electron an With

2

1

dxxq

Vdv

x

x

DC

E

∫

∫find we, )(circuit external 2

1

1

dttIVt

t DC

x

⎩⎨⎧

=s.other timeat 0

plates thebetween moving is electron thewhen /)(

dqvtI

shown as thusiscircuit external thein )(current The tI →

.2/ is (centroid)delay mean its while,/ is pulsecurrent thisof widthThe

vDvD

c =τ

me.transit ticollector the is thislayer, depletioncollector For the



)/exp( kTV0)/exp(2 ≈= kTqVnn bcpo)/exp(1 kTqVnn bepo= 2 bcpo

ofedges2at theandionsconcentratcarrierminoritytheforces and forcing diagram), (diffusion imageright the toReferring

21

cebe

nnVV

andcurrentscircuitexternalthefindmustwetheseFrom

flow. then~ and ~ currents n)(conventio diffusion Electronbase. the

ofedges2at the andionsconcentratcarrier minority theforces 21

CE

III

II

nn

.and, currentscircuit externalthefindmust we these,From BCE III

l thi hdi tithid fi d~d~ :carefully Note

II

current. alconvention of directions thein defined are and , right. left to moving electronsfor positive be to takenarebut

move.electronswhichdirectionstheindefined are and

BCE

CE

III

II



hid l ibhli l

),()(~ iscurrent convectioncollector the thenbase,

thefromregiondepletioncb theenters electron singlea If

⋅=C tqtI δ

otherwise.020 2/

)(

iscurrent terminalcollector thewhile

⎩⎨⎧ <<

= ccC

tqtI

ττ

otherwise020 2/

)(

sign opposite theof iscurrent terminalbase The

⎩⎨⎧ <<−

=

⎩

ccB

tqtI

ττ

.assumption thismakenot do we work, HBTTHzour In neglected. here be willime transit t EBbut the layer, depletion eb the toapplies analysisSimilar

otherwise.0⎩



otherwise.020 2/1

)(

pulser rectangulaa of ransform Fourier tThe

cc tth

ττ

⎩⎨⎧ <<

=

( ).)1()1()( : inorder 1 toingApproximat

.)sin()( is1st c

c

jccc

ccj

ejjjh

ejhωτ

ωτ

ωτωτωωτ

ωτωτω−−

−

≅+≅−≅

⋅=⎩

ti litt t~current terminalcollector )1(~)1(~

:currents convection the torelated be thencan currents terminalThe1

cCcCC

II

jIjII ωτωτ

≅

+⋅≅−⋅≅ −

current terminalbase )1(~current rminalemitter te

cCECEB

EE

jIIIII

II

ωτ−⋅−≅−=

≅


BJT Small-Signal Characteristics (no noise)

)/( kTV )/exp(1 kTqVnn bepo=0)/exp(2 ≈= kTqVnn bcpo

b

Rb

qAWCqAWg

2/2/2/2/

== τ

nb qADWR /=

( ) ( ))/(~

2//2//1~analysis.simplify toion)recombinat (base Ignore

1

11

bnC

bbnE

nWqADI

qAWjWqADnCjRnI

g

⋅=

⋅+⋅=+⋅=

δδ

ωδωδδ

)/()/()/()/()0(~ at DC, while,)/(But

)(

1

1

be

bnC

VkTqIVkTqWqADnWqADI

VkTqn

q

⋅=⋅⋅=⋅=→

⋅=

δδδωδ

δδ

( ) 2/ where~ and 1~ So,

)/()/()/()/()0(

2

1

nbbbeEDC

CbebEDC

E

beEDCbebnbnE

DWVkT

qIIVjkT

qII

VkTqIVkTqWqADnWqADI

=⋅=⋅+⋅=

⋅=⋅⋅=⋅=→

τδδδωτδ

δδδωδ

currents. convection electron thehavenow WekTkT



)/( kTV )/exp(1 kTqVnn bepo=0)/exp(2 ≈= kTqVnn bcpo

b

Rb

qAWCqAWg

2/2/2/2/

== τ

currentterminalcurrentconvection

nb qADWR /=

( ) ( )

bEDC

CCbEDC

C

bebEDC

EEbebEDC

E

VjqIjIIVqII

VjkT

qIIIVjkT

qII

δωτωτδδδδ

δωτδδδωτδ

⋅−⋅=−⋅≅⋅=

⋅+⋅=≅⋅+⋅=

)1()1(~~collector

1~1~emitter

currentterminalcurrentconvection

beccCCbeC VjkT

jIIVkT

I δωτωτδδδδ ≅

)1()1(collector

( ) bebEDC

becEDC

CEB VjkT

qIVjkT

qIIII δωτδωτδδδ ⋅+⋅−⋅−⋅≅−= 1 )1(

:current terminalbase

becbEDC

B VjkT

qII δττωδ ⋅+⋅= )(



wheremodel pi-hybrid standard thederivedjust have We

analysis)diffusiontheindroppingbytermlost this(we

)(

EDC

cbEDC

diff

gqIr

kTqIC ττ

=

+⋅≅

)exp(1

1)1(

analysis)diffusiontheindroppingby termlost this (we

cEDC

c

EDCc

EDCm

be

jkT

qIjkT

qIjkT

qIg

gkT

r

ωτωτ

ωτ −⋅≅+

⋅≅−⋅≅

=


BJT Noise Characteristics: Diffusion Noise

)/( kTV RbqAWg 2/2/ = τ)/exp(1 kTqVnn bepo=0)/exp(2 ≈= kTqVnn bcpo

nb

b

Rb

qADWRqAWCqAWg

/2/2/2/2/

==

τ

2

)2'(2~ 1

11

b

n

poII

WAnnqqS

grgr

⋅+⋅=

τ

)2''(2~2

)2'(2~

21

222

pob

nII

b

n

poII

nnnW

qADqS

WAnnqqS

diffdiff

grgr

++⋅⋅=

⋅+⋅=

τ

diffIzero. to voltages terminalAC set the that weimplies This

.*alone* from arising nsfluctuatiocurrent erminalconsider t We

diffEC

DCEb

npo

b

nI

III

IqnW

qADqnnnW

qADqSdiff

~~ :diagram diffusion theFrom

2'2)2''(2~,121

==

⋅=⋅⋅≅++⋅⋅=

( ) ( )

ff

fjIfjII τπτπ 2121~:currents terminalfindnow We

=−⋅=−⋅= ( ) ( )

cdiffCEBdifEE

cdiffcCC

fjIIIIIII

fjIfjII

τπ

τπτπ

2 and ~2121

⋅=−===

=−⋅=−⋅=


BJT Noise Characteristics: Diffusion Noise


nb

b

Rb

qADWRqAWCqAWg

/2/2/2/2/

==

τ

2

)2'(2~ 1

11

b

n

poII

WAnnqqS

grgr

⋅+⋅=

τ

)2''(2~2

)2'(2~

21

222

pob

nII

b

n

poII

nnnW

qADqS

WAnnqqS

diffdiff

grgr

++⋅⋅=

⋅+⋅=

τ

( )( ) ( )( ) ( )

cDCEcDCEI

IfSfS

fqIfqISC

22

,2

,

22~2~.2 inorder 1st is analysisour as 2212~ =+⋅= τπτπ

( ) ( )

DCEcIcII

DCEcIcI

qIfjSfjS

qIfSfS

diffCB

diffB

,

,22

22~2~222

⋅=⋅=

⋅=⋅=

τπτπ

τπτπ


BJT Noise Characteristics: Recombination Noise


nb

b

Rb

qADWRqAWCqAWg

/2/2/2/2/

==

τ

2

)2'(2~ 1

11

b

n

poII

WAnnqqS

grgr

⋅+⋅=

τ

)2''(2~2

)2'(2~

21

222

pob

nII

b

n

poII

n

nnnW

qADqS

WAnnqqS

diffdiff

grgr

++⋅⋅=

⋅+⋅=

τ

so0~and~~B t

)(!22

)2'(2~ 1

11⋅=

⋅+⋅=

grgr bb

n

poII

IIIII

IqWAnnqqS

τ

so,0 and , ,But

1

1

=−=

===

grCEB

CEEgrE

IIIIIIIII

2~.0~

Hence

=

=CI

qIS

S

0~2

=

=

CB

B

II

BI

S

qIS


BJT Noise Model For Circuit Analysis; Initial Result

( ) DCECIIDCECBIDEI qIfjSqIfqISqISCBBC ,,

2C, 22~ 222~ 2~

modelnoise our BJT derived have We

⋅⋅=⋅+== τπτπ

escapacitancdepletionare elementscircuit other The

indicated. as noise normal have whichs,resistancebulk andcontact escapacitanc depletion

kT


BJT Noise Model For Circuit Analysis; Reconsideration

( ) DCECIIDCECBIDEI qIfjSqIfqISqISCBBC ,,

2C, 22~ 222~ 2~

found have We

⋅⋅=⋅+== τπτπ

ime.transit tbasetoduenot but me,transit ticollector toduenoisecollector and base of ncorrelatioa found have that we troublingisIt

analysis. transportbase theon sassumption limitingour of econsequenca is This

precision.greater slightly withanalysis thedo-re usLet



.2/widthofelementsfinitethefromarisingnsexpresssio thein terms2/ theNote noise. tion/recombinageneration ignoring

noise, diffusion on econcentrat We sections. 2 into base theseparated have We

b

b

WW

.2/widthofelements finite thefrom arising bW

)/exp(1 kTqVnn bepo= 0)/exp(3 ≈= kTqVnn bcpo

/)2/(2/)2/(2/2/)2/(2/

nb

b

Rb

qADWRWqACWqAg τ

===)''(

2/2~

2111nn

WqADqS

b

nII diffdiff

+⋅⋅=

)''(2~ nnqADqS n +⋅⋅=

482)2/()2/(2/

)(2

b

n

bb

n

b

nb

DWWqA

qADWRC

q

τ==⋅=⇒

)(2/

2 3211nn

WqS

bII diffdiff

+⋅⋅=

2/)''(' 312 nnn +≅



areboxeitherthroughcurrentsdiffusionDCThe.2/ and are 2/ widthofbox theof sides 2 theon densities Electron 12 1 = nnnWb

'/2)''(2/

)''(2/

arebox either throughcurrents diffusion DCThe

E111211 ==−=−=

qADqADqAD

InW

qADnnWqADnn

WqADI

b

n

b

n

b

ndiff

ftN i

'/2)'(2/

)'(2/ E1122 ====

I

InW

qADnWqADn

WqADI

b

n

b

n

b

ndiff

)3(2/2)''(2/

2)''(2/

2~

: ofcurrent Noise

E1121

1

11⋅=+⋅⋅=+⋅⋅= Iqnn

WqADqnn

WqADqS

I

b

n

b

nII

diff

diffdiff

)1(2/2)'(2/

2)0'(2/

2~

: ofcurrent Noise

E12

2

22⋅=⋅⋅=+⋅⋅= Iqn

WqADqn

WqADqS

I

b

n

b

nII

diff

diffdiff



2/2/ that noting , from arising fluxescollector andemitter thecalculate First we 1 bdiff

RRCI τ=

[ ] [ ]8/2/14/2/12/1

2/2/1/~

2/12/

1111

1

bnnnNE

n

jIRCjIRCjRCjIRVII

RCjRIV

ωτωω

ω

+⋅=+⋅≅⎥⎦

⎤⎢⎣

⎡ +⋅=+=

+−=

[ ] [ ]

[ ] [ ]8/2/14/2/12/1

2/1/~

2/1

111

1111

bnnnC

bnnnNE

jIRCjIRCj

IRVI

jjRCj

ωτωω

ω

−⋅=−⋅≅⎥⎦

⎤⎢⎣

⎡+

⋅=−=

⎥⎦

⎢⎣ +



symmetry noting , from arising fluxescollector andemitter thecalculate thenWe 2diffI

[ ] [ ]

:functiontransfer the with thisCombining

8/2/1~ and 8/2/1~

1

22

n

bnCbnE

I

jIIjII ωτωτ +⋅=−⋅=

[ ] [ ][ ] [ ]8/2/18/2/1~

8/2/18/2/1~g

21

21

1

bnbnC

bnbnE

n

jIjII

jIjII

ωτωτ

ωτωτ

+⋅+−⋅=

−⋅++⋅=

[ ]and 2/2/))((2/2/

hence , 1~~ and ~ :analyis t timeour transi from But,

2121 nncbnnC

CCEBCC

IIjOIIIjIIIII

ττωωτ

+≅+++≅−⋅−≅≅

[ ] [ ] 2/2/4/4/

~~~

2121 CnCnbnbn

CCCEB

jIjIjIjIIjIII

ωτωτωτωτωτ

⋅+⋅+⋅−⋅=⋅+−≅



2121 4242 and 2/2/

found have We

jIjIIIII bCn

bCnBnnC ⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −⋅+⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +⋅≅+≅

ττωττω

EE

S

)1(2~ and )3(2~ where

4242

2211IqSIqS IIII diffdiffdiffdiff⋅=⋅=

⎦⎣ ⎠⎝⎦⎣ ⎠⎝

22

E

~~~

).(! 24/~ 4/~ ~:So

2211IqSSS

bCbC

IIIII diffdiffdiffdiffc

⎤⎡⎟⎞

⎜⎛⎤⎡

⎟⎞

⎜⎛

=+=

ττττ

222

E 222

4242 2211

Iq

SSS

cbbC

IIbC

IIbC

I diffdiffdiffdiffB

⎥⎥⎤

⎢⎢⎡

⎟⎟⎞

⎜⎜⎛

+⎟⎠⎞

⎜⎝⎛+⋅=

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +=

ττττω

ττωττω

E

~ 21

42~

21

42~

22

2211SjSjS

q

IIbC

IIbC

II

C

diffdiffdiffdiffcb⋅⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⋅⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +≅

⎥⎥⎦⎢

⎢⎣

⎟⎠

⎜⎝

⎟⎠

⎜⎝

ττωττω

EE 423

42 IqjIqj bCbC ⋅⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −+⋅⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +=

ττωττω



= ).(! 2~found have We

EI IqSc

⎤⎡ ⎞⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎠⎞

⎜⎝⎛+⋅+=

2222 ~ 2

22EB

cbbCI IqIqS

B

ττττω

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +⋅=

42 ~

Eb

CII jIqScb

ττω

errors.for checking warrantsand long wasderivation The


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