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6.3 BJT Modeling
N + NPIC
I B
IE
C
B
E
(a)
VBE>0
VBE<0
VBC<0
VBC>0
IC
I B
IE IF
R IR
F IF
IR
α
α C
B
E(b)
Figure 6.21
Principal Ebers-Moll Model
Injection Version
Figure 6.22 Transport Version
N + NPIC
I B
IE
C
B
E
(a)
IC
I B
IE
ICC / F
IEC
IEC / R
ICC
α α C
B
E
Injection Version
( 1)VBE
tVF ESI I e= − (6.16)
( 1)VBC
tVR CSI I e= − (6.17)
VBE>0
VBE<0
VBC<0
VBC>0
IC
I BIE IF
R IR
F IF
IR
α
α C
B
E
( 1) ( 1)VV BCBE
t tV VC F F R F ES CSI I I I e I eα α= − = − − −
( 1) ( 1)VV BCBE
t tV VE F R R ES R CSI I I I e I eα α= − + = − + −
(6.18)
(6.19)B C EI I I= − − (6.20)
Transport Version
( 1)VBE
tVCC SI I e= − (6.21)
( 1)VBC
tVEC SI I e= − (6.22)
( 1) ( 1)VV BCBE
t tV VEC SC CC S
R R
I II I I e e
α α= − = − − −
( 1) ( 1)VV BCBE
t tV VCC SE EC S
F F
I II I e I e
α α−
= + = − − + −(6.23)
(6.24)B C EI I I= − − (6.25)
IC
I BIE
ICC / F
IEC
IEC / R
ICC
α α C
B
E
IC
I B
IE
ICC / F
IEC
IEC / R
ICC
α α C
B
E
IC
IB
IE
I?1
I?2
I CT
=I CC
-I EC
C
B
E
SPICE Version
Figure 6.22
Figure 6.23
?1EC
C CC CC ECR
II I I I I
α= − = − −
?111 1 ECR
EC ECR R R
II I I
αα α β −
∴ = − = =
(6.28)
? 2CC
E EC CC ECF
II I I I I
α= − + = − + −
? 211 1 CCF
CC CCF F F
II I I
αα α β −
∴ = − = =
(6.29)
ECC CC EC
R
II I I
β= − −
Terminal currents are
CCE CC EC
F
II I I
β= − + −
B C EI I I= − −
(6.30)
where
( 1)VBE
tVCC SI I e= − (6.21)
( 1)VBC
tVEC SI I e= − (6.22)
Substitute (6.21) & (6.22) into (6.30)
1( 1) 1 ( 1)VV BCBE
t tV VC S S
R
I I e I eβ
= − − + −
(6.31)11 ( 1) ( 1)
VV BCBE
t tV VE S S
F
I I e I eβ
= − + − + −
1 1( 1) ( 1)VV BCBE
t tV VB S S
F R
I I e I eβ β
= − + −
For normal active mode
1( 1) 1 ( 1)VV BCBE
t tV VC S S
R
I I e I eβ
= − − + −
11 ( 1) ( 1)VV BCBE
t tV VE S S
F
I I e I eβ
= − + − + −
1 1( 1) ( 1)VV BCBE
t tV VB S S
F R
I I e I eβ β
= − + −
0≈
0≈
C
F
Iα
= −0≈
C
F
Iβ
=
(6.32)
Example 6.6 Ebers-Moll Model for a PNP BJT
IC
IB
IE
I CT
= I CC
-I EC
C
B
E
EC
R
Iβ
CC
F
Iβ
P +P
N
IC
IB
IE
C
B
E
1( 1) 1 ( 1)VV CBEB
t tV VC S S
R
I I e I eβ
= − − + −
11 ( 1) ( 1)VV CBEB
t tV VE S S
F
I I e I eβ
= − + − + −
1 1( 1) ( 1)VV CBEB
t tV VB S S
F R
I I e I eβ β
= − + −
1( 1) 1 ( 1)VV CBEB
t tV VC S S
R
I I e I eβ
= − − + −
11 ( 1) ( 1)VV CBEB
t tV VE S S
F
I I e I eβ
= − + − + −
1 1( 1) ( 1)VV CBEB
t tV VB S S
F R
I I e I eβ β
= − + −
0≈For normal active mode
0≈
C
F
Iα
= −
C
F
Iβ
=
0≈
Example 6.7 Ebers-Moll for Inverse Active Mode
1( 1) 1 ( 1)VV BCBE
t tV VC S S
R
I I e I eβ
= − − + −
11 ( 1) ( 1)VV BCBE
t tV VE S S
F
I I e I eβ
= − + − + −
1 1( 1) ( 1)VV BCBE
t tV VB S S
F R
I I e I eβ β
= − + −
0≈
0≈
0≈E
R
Iβ
=
(6.31)
E
R
Iα
= −
Example 6.8 Fundamental BJT Parameters
0.90353.30.72−5.00.492.6−5.00.80
IC(mA)IB(µA)VBC(V)VBE(V)
Calculate R, , and S FI β βSolution
490 188.52.6
CF
B
II
β = = =Forward current gain
900 353.3 1.55353.3
C BER
B B
I III I
β− −
= = = =
Reverse current gain
And from (6.32)
( 1)VBE
tVC SI I e= −
( 1)VBE
tVC SnI n I e = −
VBE
tVC SnI nI e= +
5 0.80(4.9 10 )0.02585
40.87
BES C
t
VnI nI n
V−∴ = − = × −
= −181.78 10 ASI −= ×
BEt
VVe≈
IC
IB
IE
C
B
E
SPICE Ebers-Moll Model
EC
R
Iβ
CC
F
Iβ
CT
CC EC
II I= −
C
B
E
Q1 3 2 1 BJTNAME.MODEL BJTNAME NPN(BF=100 CJC=20pf CJE=20pf IS=1E-16)
C B E
Substrate
CCS
P+
N +
P+
substrate emitter base collector
Second Order Effects
N + NPC
B
E
wbase
VBE0.7V VCB>>VBE
VCE VCB≅
(b)
+ +
+
≅
(a) IC
VCE VCB≅|VA |
1Co
CE o
Ig
V r∆
= =∆
slope
Early EffectEarly
Voltage
Base Width Modulation Effect
From (3.16)
2 pnS E i
base A emitter D
DDI A qn
w N w N
= +
(6.34)
C SI I∝Butincrease VCB
B C
IC
VCE VCB≅|VA |
( 0 ) ( )C BC C BC
A A BC
I V I VV V V
==
+(6.35)
Using similar triangles rule
0 0 1A BC BCS S S
A A
V V VI I I
V V +
= = +
(6.36)
Example 6.9 Early Effect
CI increases from 1.0 mA 1.1 mACEV is increased from 5.0 V 10.0 Vwhen
Calculate the Early voltage and the dynamic output resistance of this BJT.Solution
11 (0) 1 CB
C CA
VI I
V
= +
22 (0) 1 CB
C CA
VI I
V
= +
The solution can be expressed as
22 1
1
2
1
1
CCB CB
CA
C
C
IV VIV
II
−
=
−
2CE BEV V−1CE BEV V−
( )( )
9.3 4.3(1.1) 45.7 V0.1AV −= =
1 C Co
o CE CB
dI dIg
r dV dV= = ≈
The dynamic output conductance is
0(0)1 1
BEt
VCBV C
So CB A A
V Id I er dV V V
= + =
(0)A
oC
Vr
I∴ =
IC
VCE VCB≅|VA |
slope
11 (0) 1 CB
C CA
VI I
V
= +
From the measurement
1
1
1(0) 0.914 mA4.311 45.7
CC
CB
A
IIVV
∴ = = = ++
45.7 50 k(0) 0.914A
oC
Vr
I= = = Ω
Parasitic Resistances
IC
I B
IE
C
B
EBasic Model
rE
rB
rC
ICT
C
B
E
rB
rC
rE
CdC CsC
CdE CsE
CdS
DC
DE
Large Signal and Small Signal Equivalent Circuits
Figure 6.27 Large Signal Equivalent Circuit
Depletion(junction)
Stored(diffusion)
Figure 6.28 Small Signal Equivalent Circuit
ICT
C
B
E
rB
rC
rE
CdC CsC
CdE CsE
CdS
DC
DE
CB
E
rB
CoC
C rC
rE
π
µ
ππ
g mvπr
+v ro
Normal Active Mode
Example 6.10 Transconductance and input resistance
0.492.6−5.00.80IC(mA)IB(µA)VBC(V)VBE(V)
From example 6.8 calculate and mg rπ
Solution1BE BE
C t t
V VV V
m S SBE BE t
dI dg I e I edV dV V
= = =
= IC
30.49 10 18.95 mA/V0.02586
Cm
t
Ig
V
−×∴ = = =
1
CB C
BE BE FF F
m
BE
dV dVr
dIdI dI gdV
πβ
β β= = = =
188.5 9.95 k1895
F
m
rgπβ
∴ = = = Ω
CB
E
rB
CoC
C rC
rE
µ
ππ
+v ro
9.95 krπ = Ω 18.95 mAmg v vπ π=
Second Order Effects: Gummel-Poon Level in SPICEβ
ln[IC (A) ]-12 -10 -8 -6 -4
60
80
100
120
140
160
IC (A)10 -5 10 -4 10 -3 10 -2
F
Ebers-Moll model
experimental data
Figure 6.29 Current gain is a function of IC
low level
high level
1 1( 1) ( 1)VV BCBE
t tV VB S S
F R
I I e I eβ β
= − + − (6.31)
2
4 00
00 ( 1)( 1)
( 1 ( 1))
VBE
EL t
VBC
C
VBE
t
VBC
t L t
V n VS
n VS
SB
FM
VS
RM
II C I e
C I e
e
Ie
β
β
= − +
+ −
−
−+(6.37)
BE leakage emission coefficient
BC leakage emission coefficient
BE leakage saturation
current coefficient
BC leakage saturation
current coefficient
mid-current value
Gummel-Poon Charge Control Model
0
0 0( 1) ( 1)VV BCBE
t t
EBT B dE BE dC BC
C
V VB BF S R S
BT BT
AQ Q C V C V
AQ Q
I e I eQ Q
τ τ
= + +
+ − + −
(6.38)depletion charge
stored charge
A EqN A
The collector current (FA) can be expressed as
0BE
t
VVS
Cb
II e
q≈ (6.39)
0
BT
B
QQ (6.40)
Define the following parameters
0 0B BKF KR
F R
Q QI I
τ τ= =
0 0B BEA B
dC C dE
Q QAV V
C A C= =
(6.41)
(6.42)
bq can be expressed as
( )21 21 1
2
41 1 4
2 2 2b
q qq qq q
+= + ≈ + + (6.43)
where
111
1
BCBE
BCBEB A
B A
VVq
VVV VV V
= + + ≈− −
(6.44)
and
0 02 ( 1) ( 1)
VV BCBE
t tV VS S
KF KR
I Iq e e
I I= − + − (6.45)
Complete Gummel-Poon Model
02 0
04 0
( 1) ( 1)
( 1) ( 1)
V VBE BE
t EL t
V VBC BC
t CL t
V n VSB S
FM
V n VSS
RM
II e C I e
Ie C I e
β
β
= − + −
+ − + −
0
04 0( 1) ( 1)
BCBEt t
V VBC BC
t CL t
VVV VS
Cb
V n VSS
RM
II e e
qI
e C I eβ
= −
− − − −
0
02 0( 1) ( 1)
BCBEt t
V VBE BE
t EL t
VVV VS
Eb
V n VSS
FM
II e e
qI
e C I eβ
= − −
− − − −
(6.46)
GPEM
6.4 SPICE Parameters
Ebers-Moll Parameters
Gummel-Poon Parameters
Parasitic Element Related Parameters
Table 6.4
Table 6.5
Table 6.6
Parameter Measurement
VBE (V)0.0 0.2 0.4 0.6 0.8
ln[ I
C(A
)],
ln[ I B
(A)]
-40
-35
-30
-25
-20
-15
-10
-5
ln Fβ
ln IS0
IS0 = 7.7 x 10 -16 A
F = 141.2β
slope
= 1/ Vt
10 A
1 mA
µ
IC
IB
VCE = 5V
IS0 & β Measurement
VCE (V)
-80 -60 -40 -20 0 20
I C(mA
)
0
2
4
6
8
10
12
14
VCE (V)0 5 10 15
I C(m
A)
0
4
8
12IB=100 Aµ
80 Aµ
60 Aµ
40 Aµ
20 AµVA
Early Voltage Measurement
6.5 Parasitic Elements not Included in Device Models
P-type substrate
P+
N + buried layer
P+
substrate emitter base collector
P
N-epiN+
N+
E
S C
B
Parasitic substrate PNP
P-type substrate
P+ P+
substrate (V− ) isolationV +
N-epi
N+P
A BΩ4k
4pF
14pF
75
Ω
Ω
5
Parasitic capacitances in the base-diffusion resistor
P-type substrate
P+ P+
substrate (V- )
N-epi
N+P
A B
C
CP
RP
Parasitic elements in the base-collector capacitor
P-type substrate
P+ P+
substrate (V− )
N-epi
N+P
A C
CP
D
Parasitic elements in the base-collector diode
