ece 663 plans what does mcde give us for the gain? how can we use the equation to improve the gain?...
TRANSCRIPT
ECE 663
Plans
• What does MCDE give us for the gain?
• How can we use the equation to improve the gain?
• Can we develop a compact circuit model for a BJT?
ECE 663
BJT Coordinate system and parameters
ECE 663
P+ N P
nE(x’)
nE0
pB0
pB(x)
nC0
nC(x’’)
Forward Active minority carrier distribution
ECE 663
Emitter Region
• Minority carrier diffusion equation:
• Boundary conditions:
0"2
2
E
EEE
ndxnd
D
10"
0"
/0
kTqVEE
E
EBenxn
xn Wide emitter region
Law of the junction
P+ N P
nE(x’)
nE0
pB0
pB(x)
nC0
nC(x’’)
• Minority carrier diffusion equation:
• Boundary conditions:
ECE 663
Base Region
02
2
B
BBB
pdxpd
D
1
10
/0
/0
kTqVBB
kTqVBB
CB
EB
epWxp
epxpLaw of the junction(s)
P+ N P
nE(x’)
nE0
pB0
pB(x)
nC0
nC(x’’)
ECE 663
Collector Region
• Minority carrier diffusion equation:
• Boundary conditions:
0'2
2
C
CCC
n
dx
ndD
10'
0'
/0
kTqVcC
C
CBenxn
xn Wide collector region
Law of the junction
P+ N P
nE(x’)
nE0
pB0
pB(x)
nC0
nC(x’’)
ECE 663
Currents
0""
x
EEEn dx
ndqADI
0''
x
CCCn dx
ndqADI
Wx
BBCp dx
pdqADI
0
x
BBEp dx
pdqADI
ECE 663
Performance Factors and Terminal Currents
EnEp
Ep
E
Ep
II
I
I
I
Ep
CpT I
I
Tdc
dc
dcdc
1
CEB
CpCnC
EnEpE
III
III
III
ECE 663
Solutions in QN Emitter and Collector Regions
1
1)"(
)"(
/0
/"/0
/"2
/"1
kTqVE
E
EEn
LxkTqVEE
LxLxE
EB
EEB
EE
enLD
qAI
eenxn
eAeAxn
1
1)'(
)'(
/0
/'/0
/'2
/'1
kTqVC
C
CCn
LxkTqVCC
LxLxC
CB
CCB
CC
enLD
qAI
eenxn
eAeAxn
ECE 663
Solutions in the Base Region
• Need to keep both positive and negative exponential terms in the general solution.
• Apply Boundary conditions:
• Solve for A1 and A2 and plug-in to general solution
BB LxLxB eAeAxp /
2/
1)(
BBCB
EB
LWLWkTqVBB
kTqVBB
eAeAepWxp
AAepxp
/2
/1
/0
21/
0
1)(
1)0(
ECE 663
Base solutions
B
BkTqVB
B
BkTqVBB
B
BB
B
BBB
LW
Lx
ep
LW
LxW
epxp
LW
Lx
Wp
LW
LxW
pxp
CBEB
sinh
sinh
)1(
sinh
sinh
)1()(
sinh
sinh
)(
sinh
sinh
)0()(
/0
/0
0
ECE 663
Currents: Emitter hole current
0
x
BBEp dx
pdqADI
BBB LxW
LLxW
dxd
cosh1
sinh
)1(
sinh
1)1(
sinh
cosh//
0kTqV
B
kTqV
B
BB
B
BEp
CBEB e
LW
e
LW
LW
pLqAD
I
ECE 663
Collector hole current
Wx
BBCp dx
pdqADI
)1(
sinh
cosh
)1(
sinh
1 //0
kTqV
B
BkTqV
B
BB
BCp
CBEB e
LW
LW
e
LW
pLqAD
I
EC
IEp ICp
IEn ICn
IB
IE IC
ECE 663
Simplify
• Active mode biasing: VEB>0 (forward bias) and VCB<0 (reverse bias)
• Can keep only terms with emitter-base exponential
kTqVkTqV CBEB ee //
)1(
sinh
cosh/
0kTqV
B
BB
B
BEp
EBe
LW
LW
pLqAD
I
)1(
sinh
1 /0
kTqV
B
BB
BCp
EBe
LW
pLqAD
I
ECE 663
Performance factors: Emitter efficiency,
Ep
EnEnEp
Ep
E
Ep
IIII
I
I
I
1
1
)/cosh()/sinh(
)/sinh()/cosh(
1
1
0
0
/0
/0
B
B
B
E
E
B
B
E
B
BkTqVB
B
B
kTqVE
E
E
Ep
En
LWLW
pn
LL
DD
LWLW
epLqAD
enLqAD
II
EB
EB
1"
/0
0"
kTqVE
E
E
x
EEEn
EBenLD
qAdxnd
qADI
ECE 663
Emitter Efficiency
• Want to express in terms of doping:
E
i
E
iE N
npn
n2
0
2
0 B
i
B
iB N
nnn
p2
0
2
0
E
B
B
E
NN
pn
0
0
)/cosh()/sinh(
1
1
B
B
E
B
E
B
B
E
LWLW
NN
LL
DD
ECE 663
Base Transport and Common Base Gain
BEp
CpT LWI
I
/cosh1
Tdc
)1(
sinh
cosh/
0kTqV
B
BB
B
BEp
EBe
LW
LW
pLqAD
I
)1(
sinh
1 /0
kTqV
B
BB
BCp
EBe
LW
pLqAD
I
)/sinh(/cosh
1
BE
B
E
B
B
EB
dc
LWNN
LL
DD
LW
ECE 663
Common Emitter Gain
11
11
dc
dc
dcdc
1)/sinh(/cosh
1
BE
B
E
B
B
EB
dc
LWNN
LL
DD
LW
ECE 663
Can also calculate total emitter and collector currents by adding up electron and hole currents in the collector and emitter
1)1(
sinh
1)1(
sinh
cosh/
0//
0
kTqVE
E
EkTqV
B
kTqV
B
BB
B
BEnEpE
EBCBEB enLD
qAe
LW
e
LW
LW
pLqAD
III
1)1(
sinh
cosh
)1(
sinh
1 /0
//0
kTqV
XC
CkTqV
B
BkTqV
B
BB
BCnCpC
CBCBEB enLD
qAe
LW
LW
e
LW
pLqAD
III
Fortunately, for usable transistors (high gain) usually, the base is smallCompared to the minority carrier diffusion length and the equations simplify
ECE 663
Narrow Base Approximation: W<<LB
• Can simplify hyperbolic functions involving W/LB
• If <<1, then sinh() and cosh ()1 + 2/2
Wx
pWppxp
Wx
WpWx
pxp
LWLx
WpLW
LxW
pxp
BBBB
BBB
B
BB
B
BBB
)0()()0()(
)(1)0()(
//
)(/
)0()(
Linear concentration dependence across the base
ECE 663
Narrow Base Emitter Efficiency
has BB
B
B
B
LW
WlL
WlLLWLW
21)/cosh(/sinh
E
B
EB
E
BE
B
E
B
B
E
B
B
E
B
E
B
B
E
NN
LW
DD
LW
NN
LL
DD
LWLW
NN
LL
DD
1
1
1
1
)/cosh()/sinh(
1
1
If you want high emitter injection efficiency, then NB/NE << 1
High emitter doping
ECE 663
Performance factors: Base Transport factor T
2
21
1
1/cosh
1
B
BEp
CpT
LWLWI
I
If you want high base transport (T 1) then you want as small of a Base as possible W << LB or alternatively large LB = large p
ECE 663
Performance factors: Common Base Gain dc
Tdc
2
222
21
1
1
21
21
1
1
21
11
1
BE
B
EB
E
dc
BE
B
EB
E
BE
B
EB
E
BE
B
EB
E
dc
LW
NN
LW
DD
LW
NN
LW
DD
LW
NN
LW
DD
LW
NN
LW
DD
Want both high emitter doping and narrow base for high gain
ECE 663
Performance factors: Common Emitter Gain dc
11
11
dc
dc
dcdc
22
21
1
121
1
1
BE
B
EB
E
BE
B
EB
E
dc
LW
NN
LW
DD
LW
NN
LW
DD
Want both high emitter doping and narrow base for high gain
ECE 663
Circuit models
• If VCB=0 then the equation for the emitter current looks like the ideal diode equation:
)1(
sinh
cosh
)1(
sinh
cosh
1)1(
sinh
cosh
/00
000
/00
/0
/0
kTqV
FVE
B
BB
B
BE
E
EF
kTqV
B
BB
B
BE
E
EE
kTqVE
E
EkTqV
B
BB
B
BE
EB
CB
EB
EBEB
eII
LW
LW
pLD
nLD
qAI
e
LW
LW
pLD
nLD
qAI
enLD
qAe
LW
LW
pLqAD
I
ECE 663
Circuit models
If VEB=0, then the collector current equation also reduces to one that looks like an ideal diode equation:
)1(
sinh
cosh
)1(
sinh
cosh
1)1(
sinh
cosh
/00
000
/00
/0
/0
kTqV
RVC
B
BB
B
BC
C
CR
kTqV
B
BB
B
BC
C
CC
kTqVC
C
CkTqV
B
BB
B
BC
CB
EB
CB
CBCB
eII
LW
LW
pLD
nLD
qAI
e
LW
LW
pLD
nLD
qAI
enLD
qAe
LW
LW
pLqAD
I
ECE 663
Ebers-Moll Model
The exp(VCB) term in the emitter equation and the exp(VEB) term in the collector current equation have the same prefactor:
The emitter and collector current equations can be written in terms of four parameters (three are independent):
)/sinh(0
00B
B
B
BRRFF LW
pLD
qAII
)1()1(
)1()1(
/0
/0
/0
/0
kTqVR
kTqVFFC
kTqVRR
kTqVFE
CBEB
CBEB
eIeII
eIeII
Can show that F= dc
ECE 663
Ebers-Moll Equivalent Circuit – pnp BJT
ECE 663
Characteristics: Common Base
),( CBEBEE VVII ),( ECBCC IVII
Input Output
)1()1( /0
/0 kTqV
RRkTqV
FECBEB eIeII Ebers-Moll equation
)1()1( /0 kTqV
RRFEFCCBeIII After some manipulation
ECE 663
B
BB
B
BE
E
EF
LW
LW
pLD
nLD
qAI
sinh
cosh
000
B
BB
B
BC
C
CR
LW
LW
pLD
nLD
qAI
sinh
cosh
000
)/sinh(/cosh
1
BE
B
E
B
B
EB
dcF
LWNN
LL
DD
LW
0
0
00
R
FFR
RRFF
II
II
ECE 663
Common Emitter Characteristics
IE
IBIC
Input Output
),( ECEBBB VVII ),( BECCC IVII
Start with Ebers-Moll equations and some algebra to get them into the right form:
00
//00
11
11
RRFF
kTqVkTqVRRFFB
II
eeIII EBEC
00
/00
00/
00
11
11
FFR
kTqVRRFF
RRFFBkTqV
RRFFC
II
eII
IIIeIII
EC
EC
ECE 663
Common Base Characteristics
0 0.2 0.4 0.6 0.80
0.001
0.002
0.003
0.004
IEi 1
IEi 5
IEi 10
VEBi
Plots are on top of eachother for all VCB
0 10 20 30 40 500
0.001
0.002
0.003
0.004
0.005
ICi 0
ICi 1
ICi 2
ICi 3
ICi 4
ICi 5
VCB i
5mA
4mA
3mA
2mA
1mA
0mA
Input Output
ECE 663
Common Emitter Characteristics
0 0.2 0.4 0.6 0.8
0
5 10 6
1 10 5
1.510 5
IBi 0
IBi 1
IBi 2
IBi 3
IBi 4
VEBi
VEC >0VEC=0
0 20 40 60 80 1000
0.001
0.002
0.003ICi 0
ICi 1
ICi 2
ICi 3
ICi 4
VECi
IB=10
IB=7.5
IB=5
IB=2.5
IB=0
Output(Reverse biasedPN junction ..Is controlled by IB)
Input(Forward Biased PN junction)
NEW
ECE 663
Resistor-Transistor Logic (RTL)NEW