8/11/2019 derivation of ft

1/16

8/11/2019 derivation of ft

2/16

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

3/16

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

)(

8/11/2019 derivation of ft

4/16

fTwith Parasitic RSand RD

8/11/2019 derivation of ft

5/16

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

6/16

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

7/16

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

8/16

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

9/16

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

10/16

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

11/16

Derivation of fTAnd fMAX

of a BJT

8/11/2019 derivation of ft

12/16

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

13/16

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

8/11/2019 derivation of ft

14/16

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

15/16

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

8/11/2019 derivation of ft

16/16

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