Small-Signal Model
of the Forward-Active npn BJT
■
Transconductance (same concept as for MOSFET):
gm =
Ebers-Moll (forward-active):
∂i C
∂ v BE
iC = I S e
Q
v BE ⁄ V th
iC
iC
IC + ic
IC
Q
slope = gm
IC
Q
VBE
0.2
0.4
0.6 V
BE
VBE + vbe
vBE
vBE
Evaluating the derivative, we Þnd that
IC
 I S  V BE ⁄ V th
g m =  -------- e
= -------V th
 V th
EE 105 Spring 1997
Lecture 21
Input Resistance
■
The collector current is a function of the base current in the forward-active
region (recall IC = βFIB). At the operating point Q, we deÞne
∂i C
βo =
∂iB
Q
and so ic = βo ib. (Note that the ÒDC betaÓ βF and the small-signal βo are both
highly variable from device to device)
■
Since the base current is therefore a function of the base-emitter voltage, we
define the input resistance rπ as:
Ð1
rπ =
■
∂i B
∂ v BE
=
Q
∂i B
∂ iC
∂i C
Q
∂ v BE
Q
1
=  ------ g m
β 
o
Solving for the input resistance
βo
β o V th
kT β o
r π = ------ = -------------- = ------------gm
IC
qI C
■
For a high input resistance (often desirable), we need a high current gain or a low
DC bias current.
EE 105 Spring 1997
Lecture 21
Output Resistance
■
The Ebers-Moll model has perfect current source behavior in the forward-active
region -- actual characteristics show some increase:
IC
−VAn
■
VCE
Why? Base width shrinks due to encroachment by base-collector depletion
region
Approximate model: introduce Early voltage VAn to model increase in iC
Model:
■
iC = I S e
v BE ⁄ V th 
v CE 
 1 + ----------
V An

Output resistance:
∂i C
Ð1
ro =
∂ v CE
Q
IC
≅ ---------V An
EE 105 Spring 1997
Lecture 21
Numerical Values of Small-Signal Elements
ib
+
base
vbe
−
ic
collector +
+
vπ
gmvπ
rπ
ro
−
vce
−
emitter
■
Transconductance:
IC = 100 µA, Vth = 25 mV -->
gm = 4 mS = 4 x 10-3 S
Note: gm varies linearly with collector current and is independent of device
geometry, in contrast to the MOSFET
■
Input resistance:
βo = 100, IC = 100 µA, Vth = 25 mV -->
■
rπ = 25 kΩ
Output resistance:
IC = 100 µA, VAn = 35 V -->
ro = 350 kΩ
VAn = Early voltage increases with increasing base width and decreases with
decreasing base doping.
EE 105 Spring 1997
Lecture 21
,,
,,
Charge-Storage Elements:
Base-Charging Capacitance Cb
■
Minority electrons are stored in the base -- this charge qNB is a function of the
base-emitter voltage
vBE =
VBE + vbe
vBE =
VBE
dqPB
ppB(x)
NaB
base
emitter
polysilicon
contact
vBE =
VBE + vbe
negligible
hole storage
in emitter
dqNB
npB(x)
pnE(x)
−WE − xBE
■
vBE =
VBE
−xBE 0
WB
WB + xBC
x
base is still neutral... majority carriers neutralize the injected electrons
qPB = qNB
∂q PB
Cb =
∂ v BE
Q
EE 105 Spring 1997
Lecture 21
Base Transit Time
■
The electron charge in the base is found by integrating the electron concentration
in the base -- the area is AE (under the emitter):
WB
q PB = Ð q NB = Ð
∫
Ðq A E n
v BE ⁄ V th
1
( x)dx = --- q A E W B n pBo e
pB
2
0
■
The stored charge is proportional to the collector current:
2
 q A E D nB
 WB 
v BE ⁄ V th
1
q PB = --- W B ( W B ⁄ D nB )  --------------------- n pBo e
=  -------------- i C
2
 WB 
 2D nB
■
The proportionality constant looks like a diffusion time (it is) and is deÞned as
the base transit time:
2
WB
τ F = -------------2D nB
A typical transit time is τF = 10 ps for an oxide-isolated npn BJT.
■
The base-charging capacitance is:
∂q PB
Cb =
∂ v BE
= gm τ F
Q
EE 105 Spring 1997
Lecture 21
Complete Small-Signal Model
■
Add the depletion capacitance from the base-emitter junction to Þnd the total
base-emitter capacitance: Cπ = CjE + Cb
C jE =
2C jEo
CjEo is proportional to the emitter-base junction area (AE)
■
Depletion capacitance from the base-collector junction: Cµ
C µo
C µ = ------------------------------------1 + V CB ⁄ φ Bc
Cµo is proportional to the base-collector junction area (AC)
■
Depletion capacitance from collector (n+ buried layer) to bulk: Ccs
C cso
C cs = ------------------------------------1 + V CS ⁄ φ Bs
Ccso is proportional to the collector-substrate junction area (AS)
base
ib
+
rb
rc
+
Cπ
vbe
collector
ic
Cµ
vπ
rπ
gmvπ
−
ro
+
Ccs
substrate
vce
rex
−
−
emitter
EE 105 Spring 1997
Lecture 21
npn BJT SPICE model
Close correspondence to Ebers-Moll and small-signal models
Name
Parameter Description
Units
IS
transport saturation current [IS]
Amps
BF
ideal maximum forward beta [βF]
None
VAF
forward Early voltage [VAn]
Volts
BR
ideal maximum reverse beta [βR]
None
RB
zero bias base resistance [rb]
Ohms
RE
emitter resistance [rex]
Ohms
RC
collector resistance [rc]
Ohms
CJE
B-E zero-bias depletion capacitance [CjEo]
Farads
VJE
B-E built-in potential [φBe]
Volts
MJE
B-E junction exponential factor
None
CJC
B-C zero-bias depletion capacitance [Cµo]
Farads
VJC
B-C built-in potential [φBc]
Volts
MJC
B-C junction exponential factor
None
CJS
substrate zero-bias depletion capacitance [Ccso]
Farads
VJS
substrate built-in potential [φBs]
Volts
MJS
substrate junction exponential factor
None
TF
ideal forward transit time [τF]
Seconds
.MODEL MODQN NPN IS=1E-17 BF=100 VAF=25 TF=50P
+ CJE=8E-15 VJE=0.95 MJE=0.5 CJC=22E-15 VJC=0.79 MJC=0.5
+ CJS=41E-15 VJS=0.71 MJS=0.5 RB=250 RC=200 RE=5
EE 105 Spring 1997
Lecture 21
The Lateral pnp BJT
■
vertical pnp transistors cannot be made in the fabrication sequence that makes
the npn oxide-isolated transistor.
■
a ÒfreeÓ pnp can be made in which holes are injected laterally at the perimeter of
a p+ region and then diffuse across an n-type base region,where they are
collected by another p region
A
n+
(collector)
collector
p+
p
n
n+ buried
emitter
,,,
,
,,,
,,
,,,,
,,,,
,
,
,
base
p
n+ buried
layer
p+
A'
layer
p-type substrate
(a)
(emitter)
,,,
,,,,,,,,,
,
,
,
,
,,,,,,,, ,
,,,,,,,,
,
,
,
edge of n+ buried layer
(base)
,,
,,,,
,,,,
,
,
field
oxide
A
n
p+ emitter diffusion
p+
n
p
n
A
(collector)
(b)
EE 105 Spring 1997
Lecture 21
Circuit models for the Lateral pnp
■
Hole ßux in the base for the lateral pnp
holes that diffuse across
the base and are collected
n
collector
,,,
,,,,
,,,,
,
,,
,,,,
,,,,
,,,,
(p collector)
emitter
p
p+
p+
n
diffusing holes that recombine
with electrons in the base
n+ buried layer
p-type substrate
■
■
In the forward-active region, the collector current -iC is a function of vEB and the
emitter-collector voltage vEC.
v EC 
v EB ⁄ V th 
ÐiC = I S e
 1 + ----------
V Ap

The current gain is inferior: βF = 30-50; the base-transit time is more than an
order of magnitude longer ... τF = 500-700 ps
emitter
+
+
rex
veb
+
vπ
Cπ
−
base
−
vec
gmvπ
rπ
rb
−ib
ro
rc
Cbs
Cµ
−
−ic
collector
EE 105 Spring 1997
Lecture 21