ft fmax derivation
TRANSCRIPT
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Derivation of fTAnd f
MAX
of a MOSFET
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Derivation of fT
(MOSFETs)
The unity current gain frequency* (aka cutoff
frequency) Defined under the condition that the output is
loaded ith an !" short#
fT does not depend on Rgand ro
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Derivation of fT
(MOSFETs) ("ontinued)
)(2
1
)(
)()(
)(
gdgs
m
T
I
gdgsm
gdgs
m
gdgs
gdm
ins
osI
gdgsgsins
gsgdgsmgdgsmos
CC
gf
Awhen
j
CCg
CCs
g
CCs
sCg
i
iA
CCsVi
VsCVgiVgi
+
=
=
+
=+
+
=
+=
==
!ssu$e the %ero (s"gd) is s$aller co$pared to g$#
jj
CCgA T
gdgsm
I =
+
=
)(
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fT
ith &arasitic 'Sand '
D
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Derivation of fT
(MOSFETs) ("ontinued)
('Sand 'Dare included)
Millers
Approximation
[ ]
[ ])(
)(1
2
1
)(1
)(//)(
DSgd
m
gdgs
m
DSmgdgs
T
DSmgdM
DSmoDSm
gs
dV
RRCg
CC
g
RRgCC
f
RRgCC
RRgrRRgV
V
A
++
+
=
+++
=
++=
++==
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Derivation of fMAX
(MOSFETs)
fMAX* is the frequency at hich the
$ai$u$ poer gain (*aka
$ai$u$ oscillation frequency)
fMAXis defined ith
its input and output ports
con+ugate,$atched for $ai$u$poer transfer
So- e need to kno the input and
output i$pedance to define the
input and output poer as ell as
achieve the $a poer transfer$atching condition#
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Derivation of fMAX
(MOSFETs)
gdgsT
gdm
To
gdgs
gdm
o
out
gdgs
gd
t
gs
gsmdg
dggsm
o
tt
t
tout
CCC
CgCr
CC
Cg
r
Z
CC
C
V
VVgiAssume
iVgr
Vi
i
VZ
+
=
+
+
=
+
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2
osooutL
iiRR ==
For the $atching conditions-
g
sinsingS
R
ViiRZ
2===
Derivation of fMAX
(MOSFETs)("ontinued)
"on+ugate $atch at the input/ "on+ugate $atch at the output/
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&oer 0ain (1nder "on+ugate Match)
o
g
ggdT
TMAX
gdT
ogdgs
gdm
o
L
g
LTMAX
p
g
LT
g
L
ins
os
inin
outop
r
RRCf
ff
CfrCC
Cg
r
R
R
Rfff
Gwhen
R
R
f
f
R
R
i
i
Ri
RiG
+
=
+
=
++
=
==
=
=
22
1
2
1
1
1
1
2
1
1
4
1
4
122
2
21
2
21
Using thedefinition of fT
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Derivation of fTAnd f
MAX
of a BJT
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Derivation of fT
(7ipolar)
For 7ipolar Transistors-
BE
DEDE
dBC
DEdBE
mT
oegs
gdgs
d!
d"C
CCCCC
CC
gf
rVV
CCCC
=
+
+
=
)(2
"DE is due to $inority
carriers caused 3y F7
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Derivation of fT
(7ipolar) ("ontinued)
BCBEBEDE """"" +++=
QE = minority holes stored in emitterQB = minority electrons stored in base
QBE = electrons induced by the current
through the depletion region of BE-junctionQBC= electrons induced by the current
through the depletion region of BC-junction
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Derivation of fT
(7ipolar) ("ontinued)
#
m
dBCdBE
m
dBCDEdBE
mT
m#
BE
C#
BE
DEDE
s
BC
s
BE
B
B
E
EBCBEBE
C
DE#
g
CC
g
CCC
g
CC
f
gd!
di
d!
d"C
XX
D
$
D
$tttt
di
d"
+
+=
++=
+
=
==
+++=+++=
2
1
2222
22
8idth of 9eutral 'egion
8idth of
Depletion
'egion
1>
BEX
if drift current is considered#
is greater than 3ecause of reverse,3iasing#BCX
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Derivation of fT
(7ipolar)
('Sand 'Dare included)
For 3ipolar- the result is si$ilar# The only difference is that the ter$ $ust 3e
included#
[ ]
[ ]
)(
)(
)(1
2
1
)(1
)(
CEdBC#m
dBCdBE
CEdBC
m
dBC#mdBE
m
CEmdBCDEdBE
T
CEmdBCM
CEm
e
%V
RRCg
CC
RRCg
CgC
g
RRgCCC
f
RRgCC
RRgV
VA
++++
=
++++
=
++++=
++=
+=
#
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Derivation of fMAX
(7ipolar)
%
T
o
g
%T
TMAX
RC
f
r
RRCf
ff
82
2
1=
+
=
For 3ipolar transistors- there is no ter$#or