Lecture #27
OUTLINE
• BJT small signal model
• BJT cutoff frequency
• BJT transient (switching) response
Reading: Finish Chapter 12
Spring 2007
EE130 Lecture 27, Slide 1
Small-Signal Model
Common-emitter configuration,
forward-active mode:
I C   F I F 0 e qVEB / kT
B
“hybrid-pi”
BJT small signal model:
C
+
C
vbe
r
gm vbe

Transconductance:

E
E

dI C
dI C
IC
qVEB / kT
gm 

 F I F 0e

dVEB dVEB
kT / q
Spring 2007
EE130 Lecture 27, 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
forward transit time  F 
dQF

dVBE
QF
IC
 C D , BE 
Spring 2007
where QF is the magnitude
of minority-carrier charge
stored in the base and
emitter regions
d  F I C 
  F gm
dVBE
EE130 Lecture 27, Slide 3
Example: Small-Signal Model Parameters
A BJT is biased at IC = 1 mA and VCE = 3 V. dc=90, F=5 ps,
and T = 300 K. 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
12
14
c) C   F g m  5 10  0.039  1.9 10 F  19 fF (femto farad)
Spring 2007
EE130 Lecture 27, Slide 4
Cutoff Frequency, fT
ib
ib
vbe 

input admittance 1 / r   jwC
ic  g m vbe
 (w ) 
B
C
+
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:
1
 ac  1 at fT 
2  F  CJ , BE kT / qI C 
Spring 2007
EE130 Lecture 27, Slide 5
E
For the full BJT
fT 
2
equivalent circuit:
1
 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.
To maximize fT:
– increase IC
– minimize CJ,BE, CJ,BC
– minimize re, rc
– minimize F
Spring 2007
EE130 Lecture 27, Slide 6
SiGe HBT by IBM
Base Widening at High IC: the Kirk Effect
• At very high current densities (>0.5mA/mm2), base
widening occurs, so QB increases.
 F increases, fT decreases.
Top to bottom :
VCE = 0.5V, 0.8V,
1.5V, 3V.
Consider an npn BJT:
At high current levels, the density of
electrons (n  IC/qAvsat) in the
collector depletion region is
significant, resulting in widening of
the quasi-neutral base region.
As W increases, the depletion width
in the collector also increases, since
the charge density decreases:
 dep,C  qN C  qn  qN C 
IC
Avsat
At very high current densities, the
excess hole concentration in the
collector is so high that it effectively
extends the p-type base.
Spring 2007
EE130 Lecture 27, Slide 7
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
IC
gm 
kT / q 
C  C J , BE   F g m
Spring 2007
EE130 Lecture 27, Slide 8
BJT Switching - Qualitative
Spring 2007
EE130 Lecture 27, Slide 9
Turn-on transient
• We know:
dQB
Q
 I BB  B
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  Aet /  B
0  t  tr
  

t
iC (t )   t
VCC

t  tr

RL
Spring 2007
EE130 Lecture 27, Slide 10


1
t r   B ln 
 VCC / RL
1 I 
BB B







Turn-off transient
• We know:
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  

I CC 0  t  tsd


iC (t )   QB (t ) I BB B 1 e t /  B  
t
  
t
 t
Spring 2007




1



t sd   B ln 
 I CC t



I 

BB B


t
sd
EE130 Lecture 27, Slide 11
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, and also the recombination lifetime B
– By reducing B, the carrier removal rate is increased
Example: Add recombination centers (Au atoms) in the base
Spring 2007
EE130 Lecture 27, 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
Spring 2007
EE130 Lecture 27, Slide 13