Transcript Derivation of fT And fmax In Bipolar And MOSFETs

Derivation of fT And fMAX
of a MOSFET
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 Rg and ro

Derivation of fT (MOSFETs) (Continued)
i os  g m V gs  i gd  g m V gs  sC gd V gs
iins  sV gs ( C gs  C gd )
AI 
i os
iins

g m  sC gd
s ( C gs  C gd )
when A I  1
fT 
gm
2  ( C gs  C gd )
Assume the zero (sCgd) is smaller compared to gm.

gm
s ( C gs  C gd )
AI 

g m ( C gs  C gd )
j
g m ( C gs  C gd )
j

T
j
fT
with Parasitic RS and RD
Derivation of fT (MOSFETs) (Continued)
(RS and RD are included)
AV 
Vd
V gs
  g m ( R S  R D ) // ro   g m ( R S  R D )
C M  C gd 1  g m ( R S  R D ) 
Miller’s
Approximation
1
2  f T

C gs  C gd 1  g m ( R S  R D ) 
gm

C gs  C gd
gm
 C gd ( R S  R D )
Derivation of fMAX (MOSFETs)



fMAX * is the frequency at
which the maximum power
gain =1 (*aka maximum
oscillation frequency)
fMAX is defined with
its input and output ports
conjugate-matched for
maximum power transfer
So, we need to know the
input and output impedance
to define the input and
output power as well as
achieve the max power
transfer matching condition.
Derivation of fMAX (MOSFETs)
Z out 
Z in  R g 
1
j  C gs
it 
 Rg
Vt
ro
Vt
it
 g m V gs  i dg Assume i dg  g m V gs &
At high frequency (close to
fmax), we can assume that
1
j  C gs
0
So, Rg is independent of RL
Z out 
1
1
ro

g m C gd
C gs  C gd
C T ＝C gs  C gd
V gs
Vt
 ro //

C gd
C gs  C gd
CT
g m C gd
Derivation of fMAX
Conjugate match at the input:
Z S  R g iin  iins 
Vs
2Rg
(MOSFETs) (Continued)
Conjugate match at the output:
For the matching conditions,
i os
R L  R out i o 
2
Power Gain (Under Conjugate Match)
Gp 
1  i os
 
4  iins
2
o
1
2
i R out
1
2
iin R in
2
2
2
 RL
1  fT  R L



 R ＝4 f  R


g
g

when Gp 1
f  f MAX 
 RL 
1
2
fT
RL
Rg
1
1

ro
 f MAX 
1
2
Using the
definition of fT
g m C gd

C gs  C gd
1
1
ro
 2   f T  C gd
fT
2   f T  C gd R g 
Rg
ro
Derivation of fMAX
(MOSFETs)(Continued)
(RS and RD are included)
Z in  R g  R s
For high frequency condition,
Cgs → short
Hence, replace Rg by Rg+Rs
f MAX 
1
2
fT
w/ (RS+RD) term
2   f T  C gd ( R g  R s ) 
w/o (RS+RD) term
Rg  Rs
ro
Derivation of fT And fMAX
of a BJT
Derivation of fT (Bipolar)

For Bipolar Transistors,
C gs  C  C gd  C 
V gs  V be ro  
fT 
gm
2 ( C   C  )
C  ＝ C dBE  C DE
C   C dBC
C DE 
dQ DE
dv BE
CDE is due to minority
carriers caused by FB
Derivation of fT (Bipolar) (Continued)
Q DE  Q E  Q B  Q BE  Q BC
QE = minority holes stored in emitter
QB = minority electrons stored in base
QBE = electrons induced by the current
through the depletion region of BE-junction
QBC = electrons induced by the current
through the depletion region of BC-junction
Derivation of fT (Bipolar) (Continued)
Width of Neutral Region
F 
2
dQ DE
C DE 
 t E  t B  t BE  t BC 
di C
dQ DE
1
2  f T
dv BE

F
di C
dv BE
C  C 
gm

WE
2 D E
2

WB
2 D B

X BE
2 s

  F gm
C dBE  C DE  C dBC
gm

C dBE  C dBC
gm
  1 if drift current is considered.
X BC is greater than X BE because of reverse-biasing.
F
X BC
2 s
Width of
Depletion
Region
Derivation of fT (Bipolar)
(RS and RD are included)


For bipolar, the result is similar.
The only difference is that the  F term must be
included.
AV 
Vc
V be
  g m ( R E  RC )
C M  C dBC 1  g m ( R E  R C ) 
1
2  f T


C dBE  C DE  C dBC 1  g m ( R E  R C ) 
gm
C dBE  g m F  C dBC
gm

C dBE  C dBC
gm
 C dBC ( R E  R C )
  F  C dBC ( R E  R C )
Derivation of fMAX (Bipolar)
For bipolar transistors, there is no ro term.
f MAX 
1
2
fT
2   f T  C bc R b 

Rg
ro
fT
8 C bc R b