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BJT Small-Signal ModelsDr. José Ernesto Rayas Sánchez
February 19, 2007
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BJT Small-Signal Models
Dr. José Ernesto Rayas Sánchez
Some figures of this presentation were taken from the instructional resources of the following textbook:
A. S. Sedra and K. C. Smith, Microelectronic Circuits. New York, NY: Oxford University Press, 2003.
2Dr. J. E. Rayas Sánchez
Outline
DC Bias + small-signal excitation
Load lines
Amplification process in BJTs
Small-signal models
DC and small-signal analysis: example
N-port networks
Z, Y and H-parameters
Manufacturing data sheets
BJT Small-Signal ModelsDr. José Ernesto Rayas Sánchez
February 19, 2007
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3Dr. J. E. Rayas Sánchez
DC Bias + Small-Signal Excitation
If vi = 0 (bias only):0=−− BEBBBB vRiV
B
BEBBB R
vVi −= Input Load Line
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DC Bias + Small-Signal Excitation (cont)
If vi = 0 (bias only):0=−− CECCCC vRiV
C
CECCC R
vVi −= Load Line
BJT Small-Signal ModelsDr. José Ernesto Rayas Sánchez
February 19, 2007
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5Dr. J. E. Rayas Sánchez
DC Bias + Small-Signal Excitation (cont)
If vi ≠ 0
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DC Bias + Small-Signal Excitation (cont)
If vi ≠ 0
A small variation of vBE can produce a large variation in vCE
Amplification
BJT Small-Signal ModelsDr. José Ernesto Rayas Sánchez
February 19, 2007
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7Dr. J. E. Rayas Sánchez
BJT Small-Signal Models: T Model using β
Neglecting output resistance, ro
Considering ro
re
ib
B C
E
ib
re = /gm 1/gmgm = ICQ / VT
re
ib
B C
E
ib
re = /gm 1/gmgm = ICQ / VT
ro
ro = VA/ICQ
re
ib
B C
E
ib
re = /gm 1/gmgm = ICQ / VT
8Dr. J. E. Rayas Sánchez
BJT Small-Signal Models: T Model using α
Neglecting output resistance, ro
Considering ro
re
ie
B C
E
iere = /gm 1/gmgm = ICQ / VT
re
iB C
E
ire = /gm 1/gmgm = ICQ / VT
ro
ro = VA/ICQ
BJT Small-Signal ModelsDr. José Ernesto Rayas Sánchez
February 19, 2007
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9Dr. J. E. Rayas Sánchez
BJT Small-Signal Models: Hybrid π Model
Neglecting output resistance, ro
Considering ro
rgmv r
B C
E
v
gm = ICQ / VT
r gm
ro = VA/ICQ
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DC and Small-Signal Analysis – Example
100=β1=η
Bias point calculation (DC analysis)
mA023.0KΩ100
V7.0V3 =−=BI
V1.3)KΩ3(mA3.2V10 =−=CEVmA3.2100 == BC II
1-mΩ92mV25mA3.2 ===
T
CQm V
Ig
η
KΩ087.1mΩ92100
1 === −mg
r βπ
BJT Small-Signal ModelsDr. José Ernesto Rayas Sánchez
February 19, 2007
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DC and Small-Signal Analysis – Example (cont)
100=β1=η
Small-signal equivalent circuit (neglecting ro):
1-mΩ92=mgKΩ087.1=πr
12Dr. J. E. Rayas Sánchez
DC and Small-Signal Analysis – Example (cont)
Small-signal analysis
µA914.71.087KΩ100KΩV8.0 =
+=
+=
πrRvi
BB
ib
7.914 µA
BJT Small-Signal ModelsDr. José Ernesto Rayas Sánchez
February 19, 2007
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DC and Small-Signal Analysis – Example (cont)
Small-signal analysis
µA914.7=bimV6.8)µA(1.087KΩ914.7 === πriv bbe
14Dr. J. E. Rayas Sánchez
DC and Small-Signal Analysis – Example (cont)
Small-signal analysis
mV6.8=bevV374.2)mV6.8(276mV6.8)KΩ3)(mΩ92( 1 −=−=−=−= −
beCmo vRgv2.374 V
V8.0V374.2−==
i
ov v
vA
V/V96.2−=vA
BJT Small-Signal ModelsDr. José Ernesto Rayas Sánchez
February 19, 2007
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N-Ports Networks (Linear Circuits)
(R. Ludwig and P. Bretchko, RF Circuit Design, Prentice Hall, 2000)
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Impedance Matrix Representation (Z)
Each element of matrix Z is given by
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
NV
VV
M2
1
V
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
NI
II
M2
1
I
ZIV =
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
NNNN
N
N
ZZZ
ZZZZZZ
K
M
K
K
21
22221
11211
Z
jkIj
iij
kIVZ
≠=
=for 0
BJT Small-Signal ModelsDr. José Ernesto Rayas Sánchez
February 19, 2007
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17Dr. J. E. Rayas Sánchez
Admittance Matrix Representation (Y)
Each element of matrix Y is given by
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
NV
VV
M2
1
V
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
NI
II
M2
1
I
YVI =
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
=
NNNN
N
N
YYY
YYYYYY
K
M
K
K
21
22221
11211
Y
jkVj
iij
kVIY
≠=
=for 0
18Dr. J. E. Rayas Sánchez
Z-Parameters for 2-Port Networks
⎥⎦
⎤⎢⎣
⎡=
2
1
VV
V ⎥⎦
⎤⎢⎣
⎡=
2
1
II
I
ZIV =
⎥⎦
⎤⎢⎣
⎡=
2221
1211
ZZZZ
Z
Equivalent circuit:
BJT Small-Signal ModelsDr. José Ernesto Rayas Sánchez
February 19, 2007
10
19Dr. J. E. Rayas Sánchez
Y-Parameters for 2-Port Networks
⎥⎦
⎤⎢⎣
⎡=
2
1
VV
V ⎥⎦
⎤⎢⎣
⎡=
2
1
II
I
YVI =
⎥⎦
⎤⎢⎣
⎡=
2221
1211
YYYY
Y
Equivalent circuit:
20Dr. J. E. Rayas Sánchez
H-Parameters (Hybrid) for 2-Port Networks
⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
2
1
2
1
VI
IV
H
⎥⎦
⎤⎢⎣
⎡=
2221
1211
HHHH
H
Equivalent circuit:
BJT Small-Signal ModelsDr. José Ernesto Rayas Sánchez
February 19, 2007
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21Dr. J. E. Rayas Sánchez
H-Parameters for a BJT
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
2
1
2221
1211
2
1
VI
HHHH
IV
Equivalent circuit:
ib
ic
vcevbe
⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
ce
b
oefe
reie
c
eb
vi
hhhh
iv
Equivalent circuit at low freq.:
22Dr. J. E. Rayas Sánchez
BJT Manufacturing Data Sheets
BJT Small-Signal ModelsDr. José Ernesto Rayas Sánchez
February 19, 2007
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BJT Manufacturing Data Sheets (cont)
24Dr. J. E. Rayas Sánchez
BJT Manufacturing Data Sheets (cont)
BJT Small-Signal ModelsDr. José Ernesto Rayas Sánchez
February 19, 2007
13
25Dr. J. E. Rayas Sánchez
BJT Manufacturing Data Sheets (cont)
26Dr. J. E. Rayas Sánchez
BJT Manufacturing Data Sheets (cont)
BJT Small-Signal ModelsDr. José Ernesto Rayas Sánchez
February 19, 2007
14
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BJT Manufacturing Data Sheets (cont)
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Getting π Parameters from H-Parameters
Hybrid π Model: Hybrid-Parameters Model:hie
hrevce hfeibhoe vcevbe
icib
iehr ≈π
iefem hhg /≈
oeo hr /1= oeCQA hIV /=
