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
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
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
ECE594I notes, M. Rodwell, copyrighted
Bipolar Transistor AC Characteristics
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
ECE594I notes, M. Rodwell, copyrighted
Bipolar Transistor AC Characteristics
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
ECE594I notes, M. Rodwell, copyrighted
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
ECE594I notes, M. Rodwell, copyrighted
BJT Characteristics: Space Charge Transit Times
)/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
ECE594I notes, M. Rodwell, copyrighted
BJT Characteristics: Space Charge Transit Times
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⎩
ECE594I notes, M. Rodwell, copyrighted
BJT Characteristics: Space Charge Transit Times
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−⋅−≅−=
≅
ECE594I notes, M. Rodwell, copyrighted
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
ECE594I notes, M. Rodwell, copyrighted
BJT Small-Signal Characteristics (no noise)
)/( 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 ⋅+⋅= )(
ECE594I notes, M. Rodwell, copyrighted
BJT Small-Signal Characteristics (no noise)
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 −⋅≅+
⋅≅−⋅≅
=
ECE594I notes, M. Rodwell, copyrighted
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
⋅=−===
=−⋅=−⋅=
ECE594I notes, M. Rodwell, copyrighted
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
( )( ) ( )( ) ( )
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
ECE594I notes, M. Rodwell, copyrighted
BJT Noise Characteristics: Recombination 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
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
ECE594I notes, M. Rodwell, copyrighted
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
ECE594I notes, M. Rodwell, copyrighted
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
ECE594I notes, M. Rodwell, copyrighted
BJT Noise Model For Circuit Analysis; Reconsideration
.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 +≅
ECE594I notes, M. Rodwell, copyrighted
BJT Noise Model For Circuit Analysis; Reconsideration
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
ECE594I notes, M. Rodwell, copyrighted
BJT Noise Model For Circuit Analysis; Reconsideration
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
−⋅=−⋅≅⎥⎦
⎤⎢⎣
⎡+
⋅=−=
⎥⎦
⎢⎣ +
ECE594I notes, M. Rodwell, copyrighted
BJT Noise Model For Circuit Analysis; Reconsideration
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
⋅+⋅+⋅−⋅=⋅+−≅
ECE594I notes, M. Rodwell, copyrighted
BJT Noise Model For Circuit Analysis; Reconsideration
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
ECE594I notes, M. Rodwell, copyrighted
BJT Noise Model For Circuit Analysis; Reconsideration
= ).(! 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
ECE594I notes, M. Rodwell, copyrighted
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