Bipolar Junction transistor
Holes and electrons
determine device characteristics
Three terminal device
Control of two terminal currents
Amplification and switching through 3rd contact
How can we make a BJT from a pn diode?
• Take pn diode
V
p
• Remember reverse bias
characteristics
I
• Reverse saturation current: I0
n
I
I0
V
Test: Multiple choice
Why is the reverse bias current of a pn diode small?
1. Because the bias across the depletion region is
small.
2. Because the current consist of minority carriers
injected across the depletion region.
3. Because all the carriers recombine.
Test: Multiple choice
Why is the reverse bias current of a pn diode small?
1. Because the bias across the depletion region is
small.
2. Because the current consist of minority carriers
injected across the depletion region.
3. Because all the carriers recombine.
How can we make a BJT from a pn diode?
• Take pn diode
V
ep
• Remember reverse bias
characteristics
I
• Reverse saturation current: I0
Caused by minority carriers
swept across the junction
n
h+
• np and pn low
I
I0 small
I0
V
V
ep
I
Test: Multiple choice
n
h+
• If minority carrier concentration
np and/or pn
can be increased what will
happen to I0?
I
I0
V
1. Increase
2. Decrease
3. Remain the same
V
ep
• If minority carrier concentration
np and pn
I
can be increased near the
depletion region edge, then I0
will increase.
n
h+
• If np and pn higher
I
|I0| larger
I0
V
V
ep
I
Test: True-False
n
h+
I
If we only increase pn
then |I0| will still increase.
I0
V
How can we increase the minority carrier
concentration near the depletion region edge?
• Take pn diode
V
h+
p
I
n
e-
• Remember forward bias
characteristics
• How can we make a hole
injector from a pn diode?
1. By increasing the applied bias, V.
2. By increasing the doping in the p region only
3. By applying a reverse bias.
Hole injector
V
h+
p+p
• Take pn diode
I
• Remember forward bias
characteristics
• When using a p+n junction
n
e-
diode current If ≈ hole current
I
If
V
• Ip  pno (eeV/kT-1)
In  npo (eeV/kT-1)
Since NA >> ND
np << pn
→ Ip >> In
Thus:
A forward biased p+n diode is a good hole injector
A reverse biased np diode is a good minority carrier collector
V
I0
V
I
h+
p+
n
n
h+
e-
ep
If W large, then?
W
1.
2.
3.
4.
Recombination of excess holes will occur and excess will be 0 at end of layer
Recombination of excess holes will occur and excess will be large at end of layer
No recombination of excess holes will occur.
Recombination of excess electrons will occur and excess will be np0 at end of layer
Thus:
A forward biased p+n diode is a good hole injector
A reverse biased np diode is a good minority carrier collector
V
I0
V
I
h+
p+
dpn
n
e- L
p
W
n
h+ x
ep
If W large → holes
recombine
Excess hole
concentration reduces
exponentially in W to
some small value.
What is the magnitude of the hole diffusion current at the edge
x=W of the “green” region?
V
I0
V
I
h+
p+
dpn
n
e- L
p
n
h+ x
ep
W
1.
2.
3.
Magnitude of hole diffusion current at x=W is same as at x=0
Magnitude of hole diffusion current at x=W is almost 0
Magnitude of hole diffusion current cannot be derived from this layer.
Thus:
A forward biased p+n diode is a good hole injector
A reverse biased np diode is a good minority carrier collector
V
I0
V
I
h+
p+
dpn
n
e- L
p
W
Reduce W
n
h+ x
ep
if W large → holes
recombine
Since gradient of dpn @
x=W is zero, hole
diffusion current is also
zero
BJT p+np
E: emitter
B: base
C: collector
V
EB
E
IC
p+
IE
IC
B
n
VVBC
p
W < Lp
IE
VBC
C
Base: Short layer with
recombination and no Ohmic
contacts at edges.
Single junction
pno
npo
pno
npo
Double junction
npo
npo
No Ohmic contact thus minority carrier concentration not pno
How will we calculate the minority
carrier concentration in the base?
Rate equation
dp( x, t )
 2dp( x, t ) dp( x, t )
 Dp

2
t
p
x
Steady state
d 2dp( x) dp( x)
Dp

0
2
p
dx
General solution of second order differential equation
 x 
x



dp( x)  C1 exp
 C2 exp
 Lp 
 Lp 
 


With Ohmic contact
C1=0
C2≠0
Without Ohmic contact C1≠0
C2≠0
Planar BJT - npn
For integrated circuits (ICs) all contacts have to be on the top
n+-well for emitter
p+ Si
Ohmic contact B
p-well for base
E
C
n+ Si
ohmic contact
device insulation
n+ Si
p+ Si
p Si
p+ Si
n Si
p Si
p-substrate
n-well for collector
Carrier flow in BJTs
E
IE
p+
B
C
n
holes
IB
p
IC
e- gain, reverse bias
holes
IE
IC
I’B
I”B
ICB0
Recombination
e- loss
e- loss, forward bias
IB
IB = I’B + I”B – ICB0
Control by base current : ideal case.
Based upon space charge neutrality
Base region
Electrostatically neutral
IE = Ip
t transit time
t < p
h+
eWb << Lp
p
recombine with
Based on the given timescales, holes can pass through the narrow base
before a supplied electron recombines with one hole: ic/ib = p/t
The electron supply from the base contact controls the forward bias to
ensure charge neutrality!
How good is the transistor?
Injection of carriers
• Wish list:
E IEp B
C
• IEp>>IEn
IC
or g = IEp/(IEn + IEp) ≈ 1
g: emitter injection efficiency
equilibrium
• IC ≈ IEp
or B= IC/IEp ≈ 1
eVBE>0
B: base transport factor
IEn
or a= IC/IE ≈ 1
x
h + W b < Lp
a: current transfer ratio
• IB ≈ IEn + (1-B) IEp
No amplification!
thus b= IC/IB = a/(1-a)
b: current amplification factor
Amplification!
ICB0 ignored
Review 1 – BJT basics
IC
Forward active mode (ON)
IE
VBC
V
EB
E
p+
IE
IC
B
n
E
VV
BC
p
C
W < Lp
Forward biased p+n
junction is a hole injector
Reverse biased np junction is
a hole collector
Review 1 – BJT basics
IC
Forward active mode (ON)
IE
VBC
V
EB
E
p+
V
IE IB=I’B+I”B I C VBC
B
n
E
p
C
W < Lp
Forward biased p+n
junction is a hole injector
Reverse biased np junction is
a hole collector
Review 2
Amplification?
IB = I’B + I”B – ICB0
Recombination only case: I’B, ICB0 negligible
ic/ib = p/t
b = p/t
Carriers supplied by the base current stay much
longer in the base: p than the carriers supplied
by the emitter and travelling through the base: t.
But in more realistic case: I’B is not negligible
b = IC/IB
With
IB electrons supplied by base = I’B = In
IC holes collected by the collector = Ip
Currents?
• In order to calculate currents in pn junctions, knowledge of
the variation of the minority carrier concentration is
required in each layer.
• The current flowing through the base will be determined by
the excess carrier distribution in the base region.
• Simple to calculate when the short diode approximation is
used: this means linear variations of the minority carrier
distributions in all regions of the transistor. (recombination
neglected)
• Complex when recombination in the base is also taken into
account: then exponential based minority carrier
concentration in base.
Minority carrier distribution
• Assume active mode: VEB>0
& VBC<0
E
B
• Emitter injects majority carriers
into base.
dpn(0)=pno (exp(VEB/VT)-1)
C
p(x)
dp(x)
• Collector collects minority
carriers from base.
dpn(Wb)=pno (exp(VBC/VT)-1)
B
DpE
Without recombination
pn0
pn0
00
With recombination
DpC
Wb
x
Currents: simplified case
dp(x)
Assume I”B=0 & IBC0= 0
DpE
• Then IE = total current crossing
the base-emitter junction
B
• Then IC = IEp gradient of excess
hole concentration in the base
• IB without recombination is the
loss of electrons via the BE
x
junction: I’B
DpC
0
Wb
• Then IB = gradient of excess
electron concentration in the
emitter
See expressions for diode current for short diode
Narrow base: no recombination: Ip
→ minority carrier density gradient in the base
dp(x)
DpE = pn0(e eVEB/kT – 1) ≈ pn0 e eVEB/kT
DpC = pn0(e –e|VBC|/kT – 1) ≈ -pn0
DpE
Linear variation of excess carrier concentration:
dp ( x)  Ax  B
DpE  DpC
DpE
A

 Wb
Wb
B  DpE  DpC  DpE
DpC
0

x 


dp( x)  DpE 1 

 Wb 
Note: no recombination
Wb x
Collector current: Ip
Diffusion current: I p  eADp
ddp( x)
dx
ddp( x)
DpE

dx
Wb
Hole current:
DpE eAD p pn0 e
I p  eAD p

Wb
Wb
 eVEB 


 kT 
Collector current I C  I p No recombination, thus all injected
holes across the BE junction are
collected.
Base current??
Look at emitter: In
→ minority carrier density gradient in the emitter
dn(x)
Dnp = np0(e eVEB/kT – 1) ≈ np0 e eVEB/kT
Dnp
Linear variation of excess carrier concentration:
dn( x)  Ax  B
A
n p 0  Dn p
xe

Dn p
xe
B  Dn p
 Dn p 
 x  Dn p
dn( x)   
 xe 
0
xe
x
0
Base current: In
ddn( x)
Diffusion current: I n  eAD n
dx
Base current:I B
 In 
eADn n p0 e
 eVEB 


kT


xe
The base contact has to re-supply only the electrons that are
escaping from the base via the base-emitter junction since no
recombination I”B=0 and no reverse bias electron injection
into base ICB0=0.
Emitter current
The emitter current is the total current flowing through the
base emitter contact since IE=IC+IB (current continuity)
Emitter current:
 Dn n p0 D p pn0
I E  I n  I p  eA

WB
 xe
I C I p D p pn0 xe


Current gain: b 
I B I n Dn n p0 Wb

e


 eVEB 


 kT 
Short layer approach – summary
forward active mode
dc(x)
IE = IpEB + InEB
DpE
IC = IpBC + InBC
DnE
IC ≈ IpBC = IpEB
IE = IB
+ IC
IB = IE
- IC
IB = InEB
-Xe
0
DpC
DnC
Wb Xc
x
General approach also taking
recombination into account.
forward active mode
dc(x)
DpE
DnE
-Xe
-LpE
0
DpC
DnC LpC
Wb < LnB
x
Xc
Which formulae do we use for the excess minority
carrier concentration in each region?
dc(x)
forward active mode
DpE
DnE
-Xe
-LpE
0
Emitter
Collector
DpC
DnC LpC
Wb < LnB
use LONG diode approximation
dnpE(x)=DnE exp(-(-x)/LpE)
dnpC(x)=DnC exp(-x/LpC)
x
Xc
In the base we must take recombination
into account → short diode
approximation cannot be used!
dp(x)
Excess hole concentration dp(x):
d 2dp( x) dp( x)
 2
From:
2
dx
Lp
DpE
Exact solution of differential equation:
dp(x) = C1 ex/Lp + C2 e-x/Lp
Constants C1, C2:
DpE = dp(x=0)
DpC = dp(x=Wb)
DpC
Wb
x
In the base with recombination → long
diode approximation can also not be
used!
dp(x)
Exact solution of differential equation:
dp(x) = C1 ex/Lp + C2 e-x/Lp
DpE
Long diode approximation:
dp(x) = C3 e-x/Lp
Boundary condition at BC
junction cannot be guaranteed
DpC LnB
Wb
x
http://www.ecse.rpi.edu/~schubert/Course-ECSE-2210-Microelectronics-Technology-2010/
Extraction of currents in the
general approach.
forward active mode
dc(x)
IE = IpEB + InEB
DpE
IC = IpBC + InBC
IC ≈ IpBC
DnE
IE = IB
+ IC
IB = IE
- IC
-Xe
IB = InEB+ IpEB - IpBC
-LpE
0
Term due to recombination
DpC
DnC LpC
Wb < LnB
x
Xc
dp(x)
DpE
Currents: Special case when only
recombination in base current is taken
into account: Approximation: IB’=0
B
Starting point:
• Assume IE=IEp & IBC0= 0
DpC 0
Wb
• Then IE = Ip(x=0)
x
and IC = Ip(x=Wb)
• IB=IE - IC =I”B
All currents are then determined by the
minority carrier gradients in the base.
DpE = pn0(e eVEB/kT – 1)
Injection at emitter side:
dp(x)
Collection at collector side: DpC = pn0(e eVCB/kT – 1)
DpE
IE = Ip(x=0)
B
DpC
0
IC = Ip(x=Wb)
Wb
x
Expression of the diffusion currents
Diffusion current: Ip (x) = -e A Dp ddp(x)/dx
Emitter current: IE ≈ Ip (x=0)
Collector current: IC ≈ Ip (x=Wb)
Base current: IB ≈ Ip (x=0) - Ip (x=Wb)
Hyperbolic functions
IE ≈ e A Dp/Lp (DpE ctnh(Wb/Lp) - DpC csch(Wb/Lp) )
IC ≈ e A Dp/Lp (DpE csch(Wb/Lp) - DpC ctnh(Wb/Lp) )
IB ≈ e A Dp/Lp ((DpE + DpC) tanh(Wb/2Lp) )
Superposition of the effects of injection/collection at
each junction!
Note: only influence of recombination
Non-ideal effects in BJTs
• Base width modulation
V
BE
E
p+
IE
IC
Bn
VVBC
p
C
Original base width
Metallurgic junction
Effective base width
Depletion width
changes with VBC
Base width modulation
• Early voltage: VA
I c  eA
D p pn0
W
Wbb
e
 eVBE 


kT


base width modulated
iC
ideal
VA
-vCE
IB
Conclusions
• Characteristics of bipolar transistors are based on
diffusion of minority carriers in the base.
• Diffusion is based on excess carrier concentrations:
– dp(x)
• The base of the BJT is very small:
– dp(x) = C1 ex/Lp + C2 e-x/Lp
• Base width modulation changes output impedance of
BJT.
Transistor switching
Ic
t
p-type material
n-type material
iC
iC
ECC
RS
es
ib higher
RL
iB
iE
ECC /RL
ib
On
es
-vCE
ECC
t
Off
iC
iC
ic=biB
RL
iB
ECC
RS
es
Es
-Es
t
iE
-vCE
iC
ic=biB
iC
RL
ECC
RS
es
Es
-Es
t
iE
-vCE
iC
ic≠biB
iC
RL
ECC
RS
es
Es
-Es
t
Ic= ECC /RL
iE
-vCE
Switching cycle
iB
IB
iC
Switch to ON
IB≈Es/RS
Switch OFF
RL
iB
ECC
RS
DpE
QB
es
Es
-Es
-IB
DpE
iE
Qs
t
iC
DpE
t1 ts
ECC /RL
iC
IC
ECC
-vCE
t2
t’s
tsd
IC≈ECC/RL
-pn
dp
t2
ts
t1
DpC
0
t0
Wb
x
Charge in base (linear)
• Cut-off
• Saturation
– VEB<0 & VBC<0
– DpE=-pn & DpC=-pn
– VEB>0 & VBC≥0
– DpE = pn (eeVEB/kT – 1)
– DpC = 0 (VBC=0)
dp
DpE
DpE
dp
VBC>0
DpC
-pn
x
Wb
x
Wb
Currents - review.
forward active mode
dc(x)
IE = IpEB + InEB
DpE
IC = IpBC + InBC
IC ≈ IpBC
DnE
IE = IB
+ IC
IB = IE
- IC
-Xe
IB = InEB+ IpEB - IpBC
-LpE
0
Term due to recombination
DpC
DnC LpC
Wb < LnB
x
Xc
Switching cycle - review
Common emitter cicuit
iC
RL
iB
iB
IB
IB≈Es/RS
-IB
DpE
QB
es
Es
-Es
DpE
iE
Qs
t
Load line technique
iC
ECC /RL
ECC
-vCE
With IB>ICmax/b
Over-saturation
ECC
RS
Switch to ON
DpE
t1 ts
iC
IC
t2
-pno
dp
t2
ts
t1
DpC
t0
0
Wb
ICmax≈ECC/RL
pno << DpE
x
Switching cycle - review
Common emitter cicuit
iC
RL
iB
Switch OFF
-IB≈-Es/RS
ECC
RS
DpE
QB
es
Es
-Es
iB
IB
DpE
iE
Qs
t
Load line technique
iC
ECC /RL
ECC
-vCE
iC
IC
t2 t’s
tsd
t3 t4
IC≈ECC/RL
-pno
dp
t2
t’s
t3
t4
DpC
0
Wb
x
Calculating the delays
• Since the currents and minority carrier charge
storage are determined by the pn diodes, the
delays are calculated as in the pn diode.
– Knowledge of current immediately before and after
switch
– Stored minority carrier charge Qp(t) cannot change
immediately → delay.
• The additional parameter is the restriction on the
maximum collector current imposed by the load.
ON switching
OFF=0→ON
RL
C
p
RS
e(t)
B n
p
t
vbc
ECC
veb
E
iC
dpnB(x)
E I
B
IB
IB
B
ICsat
C
QB
IBp
Qsat
IB
IB
0
WB x
t<0
t≥0
veb= 0→ON≈0.7V
E-p
RS
B-n
+E>>0.7V
E
IB 
RS
tsat
i (t ) 
dQB (t ) QB (t )

dt
p


  t 
QB  I B p 1  exp 
  
 p 

&
QB 
x WB
 e A dn
x 0
tsat
t
pB
( x)dx
t<tsat
iC (t ) 
QB (t )
t
t

  t 
I B p 
1  exp 
 t 
  p 
t≥tsat
iC (t ) 
ECC
 I Csat
RL
Driving off
Time to turn the BJT OFF is determined by:
1) The degree of over-saturation (BC junction)
2) The off-switching of the emitter-base diode
ib
IB
ib
IB
CASE 1: OFF=IB=0
0N (saturation)→OFF
t
Qb
IB  p
CASE 2: OFF=-IB
0N (saturation)→OFF
t
-IB
Qb
IB  p
Qs
Qs = IC  t
iC
IC
tsd
t
t
t
-IB p
iC
IC
tsd
t
OFF switching
0N (saturation)→OFF - CASE 1: OFF=IB=0
RL
C
p
RS
e(t)
B n
p
t
vbc
ECC
veb
E
dpnB(x)
E t<0
IB
t≥0 IB=0
iC tsd
B
ICsat
C
QB
IBp
tsd
0
Qsat
WB x
t<0
veb= 0.7V (ON)→0V
E-p
RS
IB 
tsd
B-n
E=0V
E
0
RS
tsd
i (t ) 
dQB (t ) QB (t )

dt
p

 t 
QB (t )  I B p exp 
 
 p
&
QB 
tsd
t
x WB
 e A dn pB ( x)dx
x 0
t
t<tsd
iC (t ) 
ECC
 I Csat
RL
t≥tsd
iC (t ) 
QB (t )
t

I B p
t
 t 
exp 
 
 p
0N (saturation)→OFF - CASE 2: OFF=-IB
RL
C
p
RS
e(t)
B n
p
t
ECC
veb
E
E t<0
IB
t≥0
vbc
dpnB(x)
iC
tsd
B
ICsat
C
-IB
QB
IBp
tsd
0
tsd
Qsat
WB x
tsd
tsd
t
t
t<0
veb= 0.7V (ON)→-E
E-p
B-n
i (t ) 
dQB (t ) QB (t )

dt
p

RS
IB 
-E
E
RS

 t  
QB (t )  I B p 2 exp   1
  

 p 
&
QB 
x WB
 e A dn
x 0
pB
( x)dx
t<tsd
iC (t ) 
ECC
 I Csat
RL
t≥tsd
iC (t ) 
QB (t )
t

 t  
I B p 
2 exp   1
 t 
  p  
0N (saturation)→OFF - CASE 1: OFF=IB=0
iC tsd
ICsat
iC
t<tsd
E
iC (t )  CC  I Csat
RL
t≥tsd
iC (t ) 
tsd
0N (saturation)→OFF - CASE 1: OFF=-IB
QB (t )
t
tsd
iC (t ) 
ICsat

I B p
t
t<tsd
ECC
 I Csat
RL
t≥tsd
 t 
exp 
 
 p
iC (t ) 
QB (t )
t
tsd
t

 t  
I B p 
2 exp   1
 t 
  p  
t
STORAGE DELAY TIME: tsd
  t sd
ECC I B p
iC (t sd )  I Csat 

exp
 
RL
t
 p
 I B p 
t sd   p ln 

 I Csat  t 




iC (t sd )  I Csat 
ECC I B p

RL
t

 t
2 exp sd

 p




I B p


t sd   p ln

 1 1 I B p  
 
 I Csat  t  

 2 2 I Csat  t  
shorter delay
 
  1
 
 
Transients
Turn-on: off to saturation
iC
IC
IC≈ECC/RL
t
ts
Time to saturation
ON switching
QB
OFF=0→ON
IBp
RL

  t 
QB  I B p 1  exp 
  p 

 
Qsat
C
RS
e(t)
p
B n
p
t
vbc
ECC
tsat
t
veb
iC
E
iC (t ) 
ICsat
t≥tsat
t<tsat
iC (t ) 
QB (t ) I B p

t
t
ECC
 I Csat
RL

  t 
1  exp 

  p 
tsat
t=tsat
iC (t sat ) 
 t
I B p 
1  exp sat
 p
 t 

t

  I Csat


Transients
Turn-on: off to saturation
iC
IC
IC≈ECC/RL
t
ts
ts = p ln(1/( 1 – IC/b IB))
ts small when:
p small
IC small compared to b IB
oversaturation
Transients
Turn-off: saturation to off
Storage delay time: tsd
iC
IC
IC ≈ ECC/RL
toff t’s
Time from saturation
0N (saturation)→OFF - CASE 1: OFF=IB=0
iC tsd
ICsat
t<tsd
E
iC (t )  CC  I Csat
RL
t≥tsd
iC (t ) 
tsd
QB (t )
t

I B p
t
t
 t
ECC I B p

exp sd
 
RL
t
 p
 I B p 
t sd   p ln 

 I Csat  t 
iC (t sd )  I Csat 
 t 
exp 
 
 p




Transients
Turn-off: saturation to off
Storage delay time: tsd
iC
IC
IC ≈ ECC/RL
toff t’s
Determined by
EB diode
tsd = p ln(b IB /IC)
tsd small when:
p small
BUT
tsd large when:
IC small compared to b IB
NO oversaturation
Transients
Turn-on: off to saturation
iC
IC
IC≈ECC/RL
Turn-off: saturation to off
Storage delay time: tsd
iC
IC
IC ≈ ECC/RL
t
ts
ts = p ln(1/( 1 – IC/b IB))
ts small when:
p small
IC small compared to b IB
oversaturation
toff t’s
Determined by
EB diode
tsd = p ln(b IB /IC)
tsd small when:
p small
BUT
tsd large when:
IC small compared to b IB
NO oversaturation
Solution to dilemma
The Schottky diode clamp
C
C
B
B
E
E
I
B
B
metal
C
0.3 0.7
V
pn diode
Schottky diode
Large signal equivalent circuit
• Switching of BJTs
– LARGE SIGNAL
iC
RL
iB
ECC
RS
es
iE
iC
t
Ebers-Moll large signal circuit model
for large signal analysis in SPICE
Not examinable
Is valid for all bias conditions.
The excess at the BC is taken
into account what is essential
for saturation operation and offcurrents.
Superposition EB & BC influence
Take EB & BC forward biased.
Charge in base:
dp
dp
DpE
dp
DpE
DpC
=
Wb x
IEN
ICN
Wb x
+
IEI
DpC
ICI
Wb x
negative
IE = IEN + IEI
IC = ICN + ICI
Where IEN, ICI are pn diode currents of
EB and BC respectively.
Ebers-Moll equations
IE = IEN + IEI
IC = ICN + ICI
Diode currents
IE = IES (eeVEB/kT –1) – aI ICS (eeVCB/kT –1)
IC = aN IES (eeVEB/kT –1) – ICS (eeVCB/kT –1)
Ebers-Moll equations
IE = IEN + IEI
IC = ICN + ICI
Collected currents
IEI = aI ICI
ICN = aN IEN
a: current transfer factor
IE = IES (eeVEB/kT –1) – aI ICS (eeVCB/kT –1)
IC = aN IES (eeVEB/kT –1) – ICS (eeVCB/kT –1)
Ebers-Moll equations
IE = IES (eeVEB/kT –1) – aI ICS (eeVCB/kT –1)
IC = aN IES (eeVEB/kT –1) – ICS (eeVCB/kT –1)
Or:
Where: aN IES = aI ICS
IEO
eVEB/kT
eV
/kT
EB
IIEE == aaII IICC +
(1a
a
)
I
(e
+ IEO (eN I ES–1)eV /kT –1)
/kT (e CB
IICC == aaNN IIEE -- (1aCB
–1)
I) ICS–1)
ICOa(eNeV
ICO
General equivalent circuit based on diode circuit
Equivalent circuit
IE = aI IC + IEO (eeVEB/kT –1)
IC = aN IE - ICO (eeVCB/kT –1)
E
C
IC
IE
IB
B
Valid for all biasing modes
Description of different transistor regimes
• Cut-off
• Active
– VBE<0 & VCB<0
– VBE>0 & VCB<0
E
C
IE
IC
IB
E
C
IE
IC
IB
B
B
IE = -(1-aN) IES
IC = (1-aI) ICS
IC = IC0 + aN IE
IC
IE
Small!
IC0, IE=0
-VCB
BJT small signal equivalent
circuit
Now
• Amplification and maximum operation
frequency
– SMALL SIGNAL equivalent circuit
Cj,BC
B
Cj,BE
vbe
C
npn
Cd,BE
Rp
gmvbe
E
Ro
Definition of circuit elements
• Transconductance

V
I C  I 0  exp be
 VT


 


I C
IC
gm 

Vbe VT
Cj,BC
B
Cj,BE
C
Cd,BE
Rp
gmvbe
E
Ro
• Base input resistance
I C  bI B
gm 
I C
Vbe
Vbe
b
Rp 

I B
gm
Cj,BC
B
Cj,BE
C
Cd,BE
Rp
gmvbe
E
Ro
• Base-emitter input capacitances
Cd , BE 
dQb
dVbe
Cj,BE
Depletion capacitance
Cd,BE
Diffusion capacitance
See SG on pn-diode
Cd , BE
IE
 B
VT
Cj,BC
B
Cj,BE
C
Cd,BE
Rp
gmvbe
E
Ro
• Base-collector capacitance
Cj,BC
Depletion capacitance
Miller capacitance: feedback between B & C
Cj,BC
B
Cj,BE
C
Cd,BE
Rp
gmvbe
E
Ro
• Output resistance
iC
ideal
Vce  Vcb
VA
IB
-vCE
Vce VA
R0 

I C
IC
Cj,BC
B
Cj,BE
C
Cd,BE
Rp
gmvbe
E
Ro
Current gain - frequency
• Small signal current gain
ic g mvbe
b
h fe  

ib
ib
1  j C j , BE  Cd , BE Rp
Circuit analysis
ib
Cj,BC
B
Cj,BE
vbe
Max gain
C
Cd,BE
Rp
gmvbe
E
Ro
Transit frequency fT
• Small signal current gain=1
h fe  1
fT 
2p C

C

j , BE
b
j , BE
 Cd , BE
 Cd , BE Rp
b
R


p
1
2p
C j , BEVT
IE
 total transit time
 B
Base-Emitter charging time
Base transit time
Transit frequency fT
• Base transit time
Qb

 B   t 
I C 
B


IC  I p

Wb2
B 
2D
for p+n
D  D p ( pnp), Dn (npn)
Note: this approach ignores delay caused by BC junction (see 3rd year)
Simplified small signal equivalent circuit
Common-emitter connection
Active mode:
BE: forward, BC: reverse.
ib
ic
B
vbe
C
rbe
bib
rce
or
gmvbe
E
Small signal equivalent circuit when
other biasing connection is made
Common-base connection
Active mode:
BE: forward, BC: reverse.
E
ie
ic
i’e
Cdif
CjE
re
ai’e
B
CjC
rc
C
Conclusion
• Delays in BJTs are a result of the storage of
minority carriers.
• Main delay in common BJTs is due to the
base transit time t.