Transcript
Lecture 5, Slide # 1 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
Lecture 5: HBT AC properties
Lecture 5, Slide # 2 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
“Normal active” bias
• E-B forward bias (Vb>Ve)
• C-B reverse bias (Vc>Vb)
• Ice = 100 Ibe=βIbe
Ib
Ic
Ie
Like a diode.
Lecture 5, Slide # 3 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
Equivalent circuit Ic
Ib
Ie
kTeVse
BEeII /=α
B
pos XAeDn
I ≡
( ) kTeVskTeVses
BEBE eIeIII // 1)/(αα
α ≈−=→
αα
β−
=1
Lecture 5, Slide # 4 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
E-M model AC with parasitics Ic
Ib
Ie
)/( αsI
eIα Cjc
Cej
Note: at frequencies near fT, things are not quite this simple.
j is for junction (discuss) Diffusion capacitance not shown.
Lecture 5, Slide # 5 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
Global dc properties
http://www.toshiba.com/taec/components/Datasheet/2SA1244DS.pdf
It is assumed you know this, so it is rare to see on data sheets!
Note Early effect.
Lecture 5, Slide # 6 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
ac properties: notation
Ib
Ic
Ie
eEE
e
E
iIiiI
+= totalac
dc
We will use equivalent circuit #1 (implicitly).
Lecture 5, Slide # 7 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
ac properties
tibeBEBE evVv ω+=
common-emitter configuration
Rc
VCC
tibBB eiIi ω+=
tieEE eiIi ω+=
ticCC eiIi ω+=
“output”
Note: three terminal device has three-terminal equivalent ac circuit.
ticCC evVv ω+=
Lecture 5, Slide # 8 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
dc analysis
IB
IC
IE
tibeBEBE evVv ω+=
common-emitter configuration
Rc
VCC
CCCCCEC
CB
CE
kTeVSC
RIVVVIIIIeII BE
−==
=
=
=
β
α
//
/
0 for now
Lecture 5, Slide # 9 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
ac analysis
tibeBEBE evVv ω+=
Rc
VCC
tibBB eiIi ω+=
tieEE eiIi ω+=
ticCC eiIi ω+=
“output”
Note: three terminal device has three-terminal equivalent ac circuit.
ticCC evVv ω+=
We will prove:
kTeI
vi C
be
c =
“Transconductance”
Lecture 5, Slide # 10 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
Transconductance Rc
VCC
tibBB eiIi ω+=
tieEE eiIi ω+=
ticCC eiIi ω+=
ticCC evVv ω+=
tibeBEBE evVv ω+=
kTeevC
kTeevkTeVS
kTevVeS
kTevSC
tibe
tibeBE
tibeBE
BE
eI
eeI
eI
eIi
/
//
/)(
/
ω
ω
ω
=
=
=
=+
tibeBEBE evVv ω+=
x smallfor 1 xex +≈
⎟⎠
⎞⎜⎝
⎛ +≈ tibeCC e
kTevIi ω1
ticCC eiIi ω+=
But So:
kTeI
vi
kTevIi C
be
cbeCc =⇒=
Typical number is 40 mA/V.
gm
Lecture 5, Slide # 11 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
Input impedance Rc
VCC
tibBB eiIi ω+=
tieEE eiIi ω+=
ticCC eiIi ω+=
ticCC evVv ω+= tibe
CC
ticCC
B ekTevIIeiIii ω
ω
ββββ1
+=+
==tibeBEBE evVv ω+=
?=b
be
iv
tibBB eiIi ω+=
But
So:
bembe
Cb vgkTevIi
ββ==
1
So:
βm
b
be giv
=What is typical input impedance?
Lecture 5, Slide # 12 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
Gain Rc
VCC
tibBB eiIi ω+=
tieEE eiIi ω+=
ticCC eiIi ω+=
ticCC evVv ω+=
( )( )
tiCcC
tiCcCCCC
Cti
cCCC
CCCCC
eRiVeRiRIV
ReiIVRiVv
ω
ω
ω
+=
+−=
+−=
−=
tibeBEBE evVv ω+=
?=be
c
vv
But
So:
So:
Cmbe
c Rgvv
=What is typical gain?
ticCC evVv ω+=
CbemCcC RvgRiv −=−=
Lecture 5, Slide # 13 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
Summary Rc
VCC
tibBB eiIi ω+=
tieEE eiIi ω+=
ticCC eiIi ω+=
ticCC evVv ω+=ti
beBEBE evVv ω+=
bem
b vgiβ
=
beC
c vkTeIi = transcond.
input imp.
⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
=⎟⎟⎠
⎞⎜⎜⎝
⎛
c
be
C
m
c
b
vv
kTeI
g
ii
0
0β
In matrix form:
Admittance (Y) matrix
Lecture 5, Slide # 14 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
AC equivalent circuit:
Rc
VCC
tibBB eiIi ω+=
tieEE eiIi ω+=
ticCC eiIi ω+=
ticCC evVv ω+=ti
beBEBE evVv ω+=
⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
=⎟⎟⎠
⎞⎜⎜⎝
⎛
c
be
C
m
c
b
vv
kTeI
g
ii
0
0β
If we are only interested in ac components, life can be simplified:
bev
bicv
ci
Hybrid π model:
biβmgβ
Lecture 5, Slide # 15 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
T-model
Rc
VCC
tibBB eiIi ω+=
tieEE eiIi ω+=
ticCC eiIi ω+=
ticCC evVv ω+=ti
beBEBE evVv ω+=
⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
=⎟⎟⎠
⎞⎜⎜⎝
⎛
c
be
C
m
c
b
vv
kTeI
g
ii
0
0β
If we are only interested in ac components, life can be simplified:
bev
bi
cv ci
T model:
bemvg
mgα
Lecture 5, Slide # 16 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
Rules for ac analysis
• From complete circuit, calculate dc currents and voltages
• For ac analysis only: – dc voltage source -> short circuit – dc current source -> open circuit
• Replace transistor with π or T-model • Now solve (simplified) ac circuit
Lecture 5, Slide # 17 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
Next
• Generalized y-parameters • not just common emitters • Capacitances • y-parameters from doping profile • Definition of fT
Lecture 5, Slide # 18 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
AC equivalent circuit:
Rc
VCC
tibBB eiIi ω+=
tieEE eiIi ω+=
ticCC eiIi ω+=
ticCC evVv ω+=ti
beBEBE evVv ω+=
⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
=⎟⎟⎠
⎞⎜⎜⎝
⎛
c
be
C
m
c
b
vv
kTeI
g
ii
0
0β
If we are only interested in ac components, life can be simplified:
bev
bicv
ci
Hybrid π model:
biβmgβ
Discuss easy interpretation of π model.
Lecture 5, Slide # 19 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
Goal for today: Rc
VCC
tibBB eiIi ω+=
tieEE eiIi ω+=
ticCC eiIi ω+=
ticCC evVv ω+=
tibeBEBE evVv ω+= bev
bicv
ci
Hybrid π model: Red is new for ac:
been
B veigLX
M 2/
02
2
31
21 ωτ
ωω
⎥⎦
⎤⎢⎣
⎡−⎟⎟
⎠
⎞⎜⎜⎝
⎛−
m
dc
gβ
DC
jcC
jeC depC
2Me
depgC τ
≡
Lecture 5, Slide # 20 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
General admittance matrix
⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
=⎟⎟⎠
⎞⎜⎜⎝
⎛
c
be
C
m
c
b
vv
kTeI
g
ii
0
0β
Last lecture, we had emitter grounded. Called common emitter configuration:
bb vi ,
cc vi ,
ee vi ,⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
=⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
c
b
e
c
b
e
vvv
iii
...
...
...
In general:
Y-matrix has 9 elements, but once you know 4 you know them all because:
cbe iii +=and:
cebecb vvv =+ See book about details procedure to get 9 parameters from only 4.
Lecture 5, Slide # 21 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
Three configurations:
[ ] ⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛
c
be
c
b
cccb
bcbb
c
b
vv
yvv
yyyy
ii
Common emitter configuration (ve=0):
Common base configuration (vb=0):
[ ] ⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛
c
eb
c
e
ccce
ecee
c
e
vv
yvv
yyyy
ii
Common collector configuration (vc=0):
[ ] ⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛
e
bc
e
b
eeeb
bebb
e
b
vv
yvv
yyyy
ii
Easiest to calculate from doping profile.
Lecture 5, Slide # 22 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
Generalized π model: Regardless of which configuration you use, the following π model applies:
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛
2
1
2221
1211
2
1
vv
yyyy
ii
1v
1i
2v
2i11y
212vy
121vy22y
Common emitter: 1=base, 2=collector Common base: 1=emitter, 2=collector Common collector: 1=base, 2= emitter
Lecture 5, Slide # 23 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
You might be used to V=IR
Lecture 5, Slide # 24 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
General impedance matrix
bb vi ,
cc vi ,
ee vi ,
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
=⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
c
b
e
c
b
e
iii
vvv
...
...
...
Y-matrix has 9 elements, but once you know 4 you know them all because:
Lecture 5, Slide # 25 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
h matrix:
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛
2
1
2221
1211
2
1
vi
hhhh
iv
Common emitter: 1=base, 2=collector Common base: 1=emitter, 2=collector Common collector: 1=base, 2= emitter
Note: In general, matrix elements depend on dc currents, dc voltages, and frequency. Spec. sheet (or model) will provide the matrix elements as a table vs. frequency, usually for only one bias current.
Lecture 5, Slide # 26 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
Common emitter h matrix:
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛
c
b
ee
ee
c
b
vi
hhhh
iv
2221
1211
• Early effect: Collector voltage changes current gain (β).
• β depends on frequency and collector voltage. • How do we define frequency at which β = 1? • At vc=0. This is h21e
becebec ihvhihi 212221 →+=
1)(21 =Te fh• We define fT such that:
Lecture 5, Slide # 27 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
Next:
• How do we relate doping profile and geometry to Y-matrix?
• We will calculated common-base Y-matrix because it is easiest
• You can get any other Y-matrix from this.
Lecture 5, Slide # 28 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
Common base
tieEE eiIi ω+=
ticCC eiIi ω+=
ticCC evVv ω+=
tieEE evVv ω+=
cc vi ,
ee vi ,
⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
∂∂
∂∂
∂∂
∂∂
=⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛
cb
be
VCB
C
VBE
C
VCB
E
VBE
E
cb
be
ccce
ecee
c
e
vv
vi
vi
vi
vi
vv
yyyy
ii
BECB
BECB
Lecture 5, Slide # 29 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
yec
tieEE eiIi ω+=
ticCC eiIi ω+=
ticCC evVv ω+=
tieEE evVv ω+=
cc vi ,
ee vi ,
0=∂∂
=BEVCB
Eec v
iyNeglecting Early effect,
OK for GaAs HBTs, but not Si.
Lecture 5, Slide # 30 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
So far:
tieEE eiIi ω+=
ticCC eiIi ω+=
ticCC evVv ω+=
tieEE evVv ω+=
cc vi ,
ee vi ,
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛
cb
be
cb
be
ccce
ecee
c
e
vv
vv
yyyy
ii
??0?
Lecture 5, Slide # 31 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
ycc
tieEE eiIi ω+=
ticCC eiIi ω+=
ticCC evVv ω+=
tieEE evVv ω+=
cc vi ,
ee vi ,
capacitorviy
BEVCB
Ccc =
∂∂
=Neglecting Early effect, this is just capacitance of a p-n junction! (Space charge region shrinks/grows with applied voltage.)
Recall lecture 4 slide 20:
Lecture 5, Slide # 32 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
p type n type - - - - - -
+ + + + + +
p n diode capacitance
charge density no electrons or holes in depletion region
)/1(1)(2
0daa
biaap NNNq
VqANNqAXQ+
−−=−=
φε
)()(2
2 ad
ad
bi NNNN
VqA
dVdQC
+−==
φε
ap NqAXQ 0−= dn NqAXQ 0−=
dep
Cjc X
AC ε=
In Liu notation: jc stands for junction capacitance
)(2CBCB
Cdep V
qNX += φ
εwhere:
Lecture 5, Slide # 33 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
ycc
( )
jccc
VCB
Ccc
Ciy
ViCdtdVC
dtdQI
CVQ
capacitorviy
BE
ω
ω
=⇒
===
=
=∂∂
=
Lecture 5, Slide # 34 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
So far:
tieEE eiIi ω+=
ticCC eiIi ω+=
ticCC evVv ω+=
tieEE evVv ω+=
cc vi ,
ee vi ,
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟⎟
⎠
⎞⎜⎜⎝
⎛
cb
be
jccb
be
ccce
ecee
c
e
vv
Civv
yyyy
ii
ω?0?
Lecture 5, Slide # 35 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
yee
tieEE eiIi ω+=
ticCC eiIi ω+=
ticCC evVv ω+=
tieEE evVv ω+=
cc vi ,
ee vi ,
diodelikeCidiodelikecapacitorviy jc
VBE
Eee
CB
+=+=∂∂
= ω
Edep
Eje X
AC,
ε= )(2
, BEBEE
Edep VqN
X += φε
Recall lecture 5 slides 14,19 (and HW3):
Lecture 5, Slide # 36 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
“Normal active” schematic
p type base n type collector n type emitter
electrons electrons Ic Ie
carr
ier c
once
ntra
tion
x
n (no recombination in base) kTeV
ppBEenn /
0)0( =
(explain) (consider V=0)
b
kTeVp
b
p
c
Xen
eD
Xn
eD
dxxdneDJ
BE /0
)0(
)(
=
=
=Xb
(Liu 3.2)
How to calculate: Don’t use line for n(x) but tanh. dn/dx em. – dn/dx coll = base curr.
Ib
Lecture 5, Slide # 37 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
Time dependence:
p type base n type collector n type emitter
electrons electrons Ic Ie
carr
ier c
once
ntra
tion
x
n kTeVpp
BEenn /0)0( =
Ib
( ) kTevVepp
tibeBEentxn /
0),0(ω+==
0),( ≈= tXxn BpMust solve for n(x,t)! Current is dn/dx.
Lecture 5, Slide # 38 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
Time dependence:
( )
tidc
tidc
tibekTeVp
kTeVp
tibekTeVp
kTeevkTeVp
kTevVepp
exnxntxnenn
ekTevenene
kTeven
eenentxn
BEBEBE
tibeBE
tibeBE
ωω
ωω
ωω
)(~)(),( :form of is solution If )0(~)0(
1
),0(
/0
/0
/0
//0
/0
+=+→
+=⎟⎠
⎞⎜⎝
⎛ +≈
=== +
carr
ier c
once
ntra
tion
x
n kTeV
ppBEenn /
0)0( =
( ) ( ) ( )tidcti
dc
tidc
n exnxnexnxnx
Dt
exnxnnxnD
tn ωω
ω
ττ)(~)(1)(~)()(~)(
2
2
2
2
+−+∂∂
=∂+∂
⇒−∂∂
=∂∂
From lecture 2, slide 22:
)(~1)(~)(~
)(1)(0
2
2
2
2
xnxnx
Dxni
xnxnx
D dcdc
τω
τ
−∂∂
=⇒
−∂∂
=⇒
page!next see
sinh)/sinh(
)0()(
⇒
⎟⎟⎠
⎞⎜⎜⎝
⎛ −=⇒
n
B
nB
dcdc L
xXLX
nxn
(discuss)
Lecture 5, Slide # 39 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
Time dependence: ca
rrie
r con
cent
ratio
n
x
n kTeV
ppBEenn /
0)0( =
)(~1)(~)(~
)(1)(0
2
2
2
2
xnxnx
Dxni
xnxnx
D dcdc
τω
τ
−∂∂
=⇒
−∂∂
=⇒⎟⎟⎠
⎞⎜⎜⎝
⎛ −=⇒
n
B
nB
dcdc L
xXLX
nxn sinh)/sinh(
)0()(
)sinh())(sinh()0()(~
Bac
Bacebdc X
xXkTevnxn
ξξ −
=τωτ
ξDi
ac+
≡1
( ) tidctidc e
dxxndeAD
dxxdneADexnxn
dxdeAD
dxtxdneADI ωω )(~)()(~)(),(
+=+==
From lecture 2:
We only want ac part at x=0 (discuss):
⇒= tiBacac
ebdcac eX
kTevII ωξξ )coth()0( )coth()0( Bacacdc
eb
acee X
kTeI
vIy ξξ=≡
Lecture 5, Slide # 40 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
Time dependence: kTeV
ppBEenn /
0)0( =
carr
ier c
once
ntra
tion
x
n
)coth()0( Bacacdceb
acee X
kTeI
vIy ξξ=≡
2021
Btransit XD
=≡τ
ω
For small frequencies,
DeBacacdc CigXkTeI ωξξ +≈)coth()0(
DXgC B
eD 3
2
≡
The “diffusion” capacitance is the extra (time dependent) charge stored as excess electrons in the base:
kTeIg dce )0(≡
Lecture 5, Slide # 41 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
So far:
tieEE eiIi ω+=
ticCC eiIi ω+=
ticCC evVv ω+=
tieEE evVv ω+=
cc vi ,
ee vi ,
⎟⎟⎠
⎞⎜⎜⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛ ++=⎟⎟
⎠
⎞⎜⎜⎝
⎛
cb
be
jc
jeDe
c
e
vv
CiCiCig
ii
ω
ωω
?0
Lecture 5, Slide # 42 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
yce
tieEE eiIi ω+=
ticCC eiIi ω+=
ticCC evVv ω+=
tieEE evVv ω+=
cc vi ,
ee vi ,
?=∂∂
=CBVBE
Cce v
iy
We need solution of time-dependent diffusion equation again. ic equal to dn/dx at collector side.
Lecture 5, Slide # 43 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
Time dependence: ca
rrie
r con
cent
ratio
n
x
n kTeV
ppBEenn /
0)0( =
)(~1)(~)(~
)(1)(0
2
2
2
2
xnxnx
Dxni
xnxnx
D dcdc
τω
τ
−∂∂
=⇒
−∂∂
=⇒
)sinh())(sinh()0()(~
Bac
Bacebdc X
xXkTevnxn
ξξ −
=τωτ
ξDi
ac+
≡1
( ) tidctidc e
dxxndeAD
dxxdneADexnxn
dxdeAD
dxtxdneADI ωω )(~)()(~)(),(
+=+==
From lecture 2:
We only want ac part at x=XB (discuss):
⇒⎟⎟⎠
⎞⎜⎜⎝
⎛−= )(csc2
1 2
2
Bacacen
Bebac Xhg
LXvI ξξ )(csc
21 2
2
Bacacen
B
eb
acce Xhg
LX
vIy ξξ⎟⎟
⎠
⎞⎜⎜⎝
⎛−≡
not quite…
Lecture 5, Slide # 44 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
Time dependence: ca
rrie
r con
cent
ratio
n
x
n kTeV
ppBEenn /
0)0( =
2/2
2
2
2
)(csc2
1
)(csc2
1
MiBacace
n
B
Bacacen
B
eb
acce
eXhgLX
XhgLX
vIy
ωτξξ
ξξ
−
⎟⎟⎠
⎞⎜⎜⎝
⎛−→
⎟⎟⎠
⎞⎜⎜⎝
⎛−=≡
It takes an electron some time to get through depletion region here. (Question for class: What about the other depletion region?)
sat
depM v
X≡τ This is how long it takes an electron to get through depletion region.
(There is no scattering in the depletion region in this approximation.)
2/
02
2
31
21 Mi
en
Bce eig
LXy ωτ
ωω −
⎥⎦
⎤⎢⎣
⎡−⎟⎟
⎠
⎞⎜⎜⎝
⎛−−≈
At low frequencies:
Lecture 5, Slide # 45 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
So far:
tieEE eiIi ω+=
ticCC eiIi ω+=
ticCC evVv ω+=
tieEE evVv ω+=
cc vi ,
ee vi ,
⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
⎥⎦
⎤⎢⎣
⎡−⎟⎟
⎠
⎞⎜⎜⎝
⎛−−≈
++
=⎟⎟⎠
⎞⎜⎜⎝
⎛−
cb
be
jci
en
Bce
jeDe
c
e
vv
CieigLXy
CiCig
ii
M ωωω
ωω
ωτ 2/
02
2
31
21
0
You will calculate for typical parameters in HW.
Lecture 5, Slide # 46 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
Next steps
• Translate back to common-emitter model • Generate equivalent circuit.
Lecture 5, Slide # 47 EECS277B © P.J. Burke, Fall 2013 Last modified 9/19/13 19:56
AC equivalent circuit: Rc
VCC
tibBB eiIi ω+=
tieEE eiIi ω+=
ticCC eiIi ω+=
ticCC evVv ω+=
tibeBEBE evVv ω+= bev
bicv
ci
Hybrid π model: Red is new for ac:
been
B veigLX
M 2/
02
2
31
21 ωτ
ωω
⎥⎦
⎤⎢⎣
⎡−⎟⎟
⎠
⎞⎜⎜⎝
⎛−
m
dc
gβ
DC
jcC
jeC depC
2Me
depgC τ
≡
• Circuit model good only for low frequencies • At high frequencies computer must be used! • That concludes our derivation of intrinsic HBT behavior. • Next will include parasitics, and discuss fT, fmax
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