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Unformatted text preview: eVBE /UT − eVBC /UT where 2 JS ≡ BJT qni WB pb
dx
0 Dn 211 Analog ICs; JiehTsorng Wu Gummel Number (G)
Dn is a weak function of x . Then, JS can be expressed as
2 JS = where
G≡ WB qni WB pb
dx
0 Dn pb(x )dx ≈ 0 2 qni D n = G WB NA(x )dx
0 • The Gummel number, G , is simply the dopant concentration per unit crosssectional
area of the base.
• For a uniform base region, NA(x ) = NA, then G = WB NA. BJT 212 Analog ICs; JiehTsorng Wu Base Transport Current
The total minority carrier transport current across the base is
2 IT = JN × A = IS eVBE /UT − eVBC /UT where IS = JS × A = qni D n
G ×A The transport current can be separated into forward and reverse components as
IT = IS eVBE /UT − 1 − IS eVBC /UT − 1 = ICF + IE R
• If VBE > 0 and VBC < 0, the device is biased in the forwardactive region,
IT = IS eVBE /UT
• If VBE < 0 and VBC > 0, the device is biased in the inverseactive region,
IT = IS eVBC /UT
• If VBE > 0 and VBC > 0, the device is biased in the saturation region.
BJT 213 Analog ICs; JiehTsorng Wu Base Current
In the forwardactive region
IB = IBB + IBE
• IBB is due to the recombination of holes and electrons in the base.
• IBE is due to the injection of holes from the base into the emitter.
Deﬁne Qe as the minority carrier charge in the base region
Qe = qA 2 WB nb(x )dx
0 or ni V /U
1
1
Qe = qAWB nb(0) = qAWB e BE T
2
2
NA IBB is related to Qe by the lifetime of minority carriers in the base, τb
IBB BJT 2
Qe 1 qAWB ni
=
=
· eVBE /UT
τb
2 τb NA 214 Analog ICs; JiehTsorng Wu Base Current
IBE depends on the gradient of minority carriers (holes) in the emitter.
• For a “longbase” emitter (all minority carriers recombine in the quasineutral region)
with a diﬀusion length Lp
IBE = qADp
Lp peoeVBE /UT = qADp n2
i
Lp ND eVBE /UT ND = Emitter Doner Density • For a “shortbase” emitter (all recombination at the contact) with emitter width WE , WE
simply replaces Lp in the expression for IBE .
The total base current in the forwardactive region is
2
qADp n2
1 qAWB ni
i
eVBE /UT
+
IB =
2 τB NA
Lp ND • In modern narrowbase transistors IBE
BJT IBB .
215 Analog ICs; JiehTsorng Wu Forward Current Gain
In the forwardactive region, the forward current gain is
βF ≡ IC
IB 1 = 2
WB DWN 2τb Dn + Dp L B N A
n P D The emitter current is
IE = −(IC + IB ) = − IC +
where IC
=−
αF IC
βF IC βF
1
=
=
αF ≡ − =
IE βF + 1 1 + 1
β
F αT = 1
2
WB 1 + 2τ 1
2
WB 1 + 2τ
γ= b Dn +
1 ≈ αT · γ DWN 1 + Dp L B N A
n B Dn Dp WB NA
Dn LP ND P D • αT is called the base transport factor, and γ is called the emitter injection eﬃciency.
BJT 216 Analog ICs; JiehTsorng Wu BJT DC LargeSignal Model in ForwardActive Region
IB IB IC
B C B IC VBE C VBE(on)
IE IE
E E IS V /U
IB =
e BE T
βF IC = βF IB • The voltage on the emitter junction can be approximated by a constant VBE (on) .
◦ • VBE (on) is usually 0.6 V to 0.8 V, and has a temperature coeﬃcient of −2 mV/ C. BJT 217 Analog ICs; JiehTsorng Wu Dependence of βF on Operating Condition • At high currents, due to highlevel injection
IC → IS eVBE /(2UT )
• At low currents, due to recombination in the BE depletion region
IB → IS eVBE /(2UT )
BJT 218 Analog ICs; JiehTsorng Wu Collector Voltage Eﬀects In the forwardactive region, an increase ∆VCE in VCE results in an increase in the
collector depletion layer width, thereby reducing WB by ∆WB , and increasing IC .
2 IC = IS eVBE /UT = A
∂IC
∂VCE
BJT qni Dn
G eVBE /UT 2 = −A qni Dn
G2 e VBE /UT 219 G = Gummel number
IC d G
dG
·
=− ·
d VCE
G d VCE
Analog ICs; JiehTsorng Wu Collector Voltage Eﬀects
For a uniformbase transistor
G = WB NA and ∂IC
∂VCE IC d WB
=−
·
WB d VCE • d WB /d VCE is typically a weak function of VCE for a reverse biased collector junction
and is often assumed to be constant.
The Early voltage, VA, is given by
VA = IC
∂IC /∂VCE = −WB 1
d WB /d VCE The inﬂuence of changes in VCE on IC can thus be represented as
IC = IS e VBE /UT 1+ VCE
VA • Typical values of VA are 15–100 V.
BJT 220 Analog ICs; JiehTsorng Wu Base Transport Model
C IC IS /βR IT B IE IS /βF
E IT = IS eVBE /UT − eVBC /UT
IC = IT − BJT IS
IS
IE = −IT −
eVBC /UT − 1
eVBE /UT − 1
βR
βF
IS
IS
IB =
eVBE /UT − 1 +
eVBC /UT − 1
βF
βR 221 Analog ICs; JiehTsorng Wu EbersMoll Model
Recalling
IT = IS eVBE /UT − eVBC /UT
IC = IT − IS eVBC /UT − 1 IE = −IT − IS eVBE /UT − 1 βR
βF
SPICE uses the base transport model with the equations rewritten as:
IC = IS e VBE /UT − 1 − IS IE = −IS 1
1+
βF e VBE /UT 1
1+
βR e VBC /UT − 1 −IS e VBC /UT − 1 = IS e VBE /UT IS
−1 −
eVBC /UT − 1
αR IS
−1 =−
eVBE /UT − 1 −IS eVBC /UT − 1
αF • Note that, in the classical EbersMoll model, parameters IE S and ICS are deﬁned such
that
αF IE S = αR ICS = IS BJT 222 Analog ICs; JiehTsorng Wu Leakage Current
VBE /UT In the forwardactive region, e
IC ≈ IS e VBE /UT 1 and e
IS
+
αR VBC...
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 Winter '09
 Choma
 Integrated Circuit, Transistor, The Land

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