8/11/2019 derivation of ft
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8/11/2019 derivation of ft
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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 with an AC short.
fT does not depend on Rgand ro
8/11/2019 derivation of ft
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Derivation of fT
(MOSFETs) (Continued)
)(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
Assume the zero (sCgd) is smaller compared to gm.
jj
CCgA T
gdgsm
I
)(
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fTwith Parasitic RSand RD
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Derivation of fT
(MOSFETs) (Continued)
(RSand RDare 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
8/11/2019 derivation of ft
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Derivation of fMAX
(MOSFETs)
fMAX
* is the frequency atwhich the maximum power
gain =1 (*aka maximum
oscillation frequency)
fMAXis defined withits input and output ports
conjugate-matched for
maximum power transfer
So, we need to know theinput and output impedance
to define the input and
output power as well as
achieve the max power
transfer matching condition.
8/11/2019 derivation of ft
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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
//1
1
&
g
gs
gin RCj
RZ
1
At high frequency (close tofmax), we can assume that
So, Rg is independent of RL
01
gsCj
8/11/2019 derivation of ft
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2
osooutL
iiRR
For the matching conditions,
g
sinsingS
R
ViiRZ
2
Derivation of fMAX
(MOSFETs)(Continued)
Conjugate match at the input: Conjugate match at the output:
8/11/2019 derivation of ft
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Power Gain (Under Conjugate 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
8/11/2019 derivation of ft
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Derivation of fMAX
(MOSFETs)(Continued)
(RSand RDare included)
sgin RRZ
Hence, replace Rgby Rg+Rs
o
sg
sggdT
TMAX
r
RRRRCf
ff
)(22
1
For high frequency condition,
Cgs short
w/o (RS+RD) term
w/ (RS+RD) term
8/11/2019 derivation of ft
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Derivation of fTAnd fMAX
of a BJT
8/11/2019 derivation of ft
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Derivation of fT
(Bipolar)
For Bipolar Transistors,
BE
DEDE
dBC
DEdBE
mT
obegs
gdgs
dv
dQC
CCCCC
CC
gf
rVV
CCCC
)(2
CDE is due to minority
carriers caused by FB
8/11/2019 derivation of ft
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Derivation of fT
(Bipolar) (Continued)
BCBEBEDE QQQQQ
QE = minority holes stored in emitterQB = minority electrons stored in baseQBE = 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
(Bipolar) (Continued)
F
m
dBCdBE
m
dBCDEdBE
mT
mF
BE
CF
BE
DEDE
s
BC
s
BE
B
B
E
EBCBEBE
C
DEF
g
CC
g
CCC
g
CC
f
gdv
di
dv
dQC
XX
D
W
D
Wtttt
di
dQ
2
1
2222
22
Width of Neutral Region
Width of
Depletion
Region
1
BEX
if drift current is considered.
is greater than because of reverse-biasing.BCX
8/11/2019 derivation of ft
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Derivation of fT
(Bipolar)
(RSand RDare included)
For bipolar, the result is similar.
The only difference is that the term must be
included.
)(
)(
)(1
2
1
)(1
)(
CEdBCFm
dBCdBE
CEdBC
m
dBCFmdBE
m
CEmdBCDEdBE
T
CEmdBCM
CEm
be
cV
RRCg
CC
RRCg
CgC
g
RRgCCC
f
RRgCC
RRgV
VA
F
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Derivation of fMAX
(Bipolar)
bbc
T
o
g
bbcT
TMAX
RC
f
r
RRCf
ff
82
2
1
For bipolar transistors, there is no term.or
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