# Lecture 28 OUTLINE The BJT (cont’d) • Small-signal model

Lecture 28
OUTLINE
The BJT (cont’d)
• Small-signal model
• Cutoff frequency
• Transient (switching) response
Small-Signal Model
Common-emitter
configuration,
forward-active mode:
I C   F I F 0 e qVBE / kT
R. F. Pierret, Semiconductor Device Fundamentals, Fig.12.1(a)
B
C
+
“hybrid pi”
BJT small signal model:
C
vbe

E

dI C
IC
d
qVBE / kT
gm 

 F I F 0e

dVBE dVBE
kT / q
EE130/230A Fall 2013
gm vbe

E
Transconductance:
r
Lecture 28, Slide 2
Small-Signal Model (cont.)
gm
1
dI B
1 dI C



r dVBE  dc dVBE  dc

r 
 dc
gm
C  C J , BE  C D , BE  C D , BE
CJ, BE
A s

Wdep, BE
CD,BE
dQF

dVBE
QF
forward transit time  F 
IC
 C D , BE
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where QF is the magnitude of
minority-carrier charge stored in
the base and emitter regions
d  F I C 

  F gm
dVBE
Lecture 28, Slide 3
Example
A BJT is biased at IC = 1 mA and VCE = 3V. dc = 90, F = 5ps, T = 300K.
Find (a) gm , (b) r , (c) C .
Solution:
(a)
g m  I C /( kT / q) 
1 mA
mA
 39
 39 mS (milli siemens)
26 mV
V
(b) r = dc / gm = 90/0.039 = 2.3 kW
(c) C   F g m  5 10 12  0.039  1.9 10 14 F  19 fF (femto farad)
EE130/230A Fall 2013
Lecture 28, Slide 4
Cutoff Frequency, fT
B
ib
ib
vbe 

input admittance 1 / r   jwC
+
C
ic  g m vbe
 (w ) 
C
vbe
r
gm vbe

E
ic
gm
gm


1 / r   jwC g m /  dc   jw  F g m  C J , BE 
ib
1

1 /  dc   jw  F  C J , BE kT / qI C 
The cutoff frequency is defined to be the frequency (f = w/2)
at which the short-circuit a.c. current gain equals 1:
 ac  1 at fT 
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1
2  F  CJ , BE kT / qI C 
Lecture 28, Slide 5
E
For the full BJT
equivalent circuit:
1
fT 
2  F  CJ , BE  CJ , BC kT / qI C  CJ , BC re  rc 
fT is commonly used
as a metric for the
speed of a BJT.
Si/SiGe HBT by IBM
To maximize fT:
• increase IC
• minimize CJ,BE, CJ,BC
• minimize re, rc
• minimize F
EE130/230A Fall 2013
Lecture 28, Slide 6
Base Widening at High IC: Kirk Effect
For a NPN BJT:
J C  qnvsat   dep,C  qN C  qn  qN C 
JC
vsat
• At very high current densities (&gt;0.5mA/mm2), the density of mobile charge passing
through the collector depletion region exceeds the ionized dopant charge density:
increasing IC
 The base width (W) is effectively increased (referred to as “base push out”)
 F increases and hence fT decreases.
•This effect can be avoided by increasing NC  increased CJ,BC , decreased VCE0
EE130/230A Fall 2013
Lecture 28, Slide 7
C. C. Hu, Modern Semiconductor Devices for Integrated Circuits, Figure 8-18
Summary: BJT Small Signal Model
Hybrid pi model for the common-emitter configuration,
forward-active mode:
B
C
+
C
vbe
r
gm vbe

E
E
r 
 dc
gm
gm 
C  C J , BE   F g m
EE130/230A Fall 2013
Lecture 28, Slide 8
IC
kT / q 
BJT Switching - Qualitative
R. F. Pierret, Semiconductor Device Fundamentals, Figs. 12.3-12.4
EE130/230A Fall 2013
Lecture 28, Slide 9
Turn-on Transient Response
dQB
QB
 I BB 
dt
B
where IBB=VS/RS
•The general solution is:
QB (t )  I BB B  Aet /  B
•Initial condition: QB(0)=0
since transistor is in cutoff
QB (t )  I BB B (1  e

t /  B
)

 QB (t ) I BB B 1  e t / B

0  t  tr

 
t
iC (t )   t
VCC

t  t r

RL

EE130/230A Fall 2013
Lecture 28, Slide 10


1
t r   B ln 
 VCC / RL
1 I 
BB B







R. F. Pierret, Semiconductor Device Fundamentals, Fig. 12.5
Turn-off Transient Response
dQB
Q
 I BB  B
dt
B
• The general solution is:
QB (t )  I BB B  Aet /  B
• Initial condition: QB(0)=IBBB

QB (t )  I BB B 1   et /  B  





1



t sd   B ln 
 I CC t



I 

 BB B

I CC  0  t  t sd



iC (t )  
t /  B


Q
(
t
)
I

1


e

 B  BB B
 t  t sd
  t
t

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
Lecture 28, Slide 11
R. F. Pierret, Semiconductor Device Fundamentals, Fig. 12.5
Reducing B for Faster Turn-Off
• The speed at which a BJT is turned off is dependent on the
amount of excess minority-carrier charge stored in the base,
QB, and also the recombination lifetime, B.
– By reducing B, the carrier removal rate is increased
Example: Add recombination centers (Au atoms) in the base
EE130/230A Fall 2013
Lecture 28, Slide 12
Schottky-Clamped BJT
• When the BJT enters the saturation mode, the Schottky
diode begins to conduct and “clamps” the C-B junction
voltage at a relatively low positive value.
 reduced stored charge in quasi-neutral base
EE130/230A Fall 2013
Lecture 28, Slide 13
R. F. Pierret, Semiconductor Device Fundamentals, Fig. 12.7