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CONSTRAINTS IN THE DESIGN OF GALLIUM-ARSENIDE MESFET MMIC DISTRIBUTED AMPLIFIERS (MICROWAVE, MONALYTHIC, TRAVELING WAVE)

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R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r re p r o d u c tio n prohib ited w ith o u t p e r m is s io n .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
8516755
Becker, Robert Charles
CONSTRAINTS IN THE DESIGN OF GALLIUM-ARSENIDE MESFET MMIC
DISTRIBUTED AMPLIFIERS
The University o f Wisconsin ■Madison
University
Microfilms
International
Ph.D.
1985
300 N. Z eeb Road, Ann Arbor, Ml 48106
Copyright 1985
by
Becker, Robert Charles
All Rights Reserved
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
CONSTRAINTS IN THE DESIGN OF
GaAs MESFET MMIC
DISTRIBUTED AMPLIFIERS
A thesis submitted to the Graduate School of the
University of Wisconsin-Madison in partial fulfillment of
the requirements for the degree of Doctor of Philosophy
by
Robert Charles Becker____________
Degree to be awarded:
December 19_____
May 19 _____
August 1985
Approved by Thesis Reading Committee:
29 April,
1985
Date of Examination
Mayor Professor
/
•2 h _ i
y
Dean, Graduate School
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
CONSTRAINTS IN THE DESIGN OF
GaAs MESFET MMIC
DISTRIBUTED AMPLIFIERS
by
Robert Charles Becker
A Thesis submitted in partial fulfilment of the
requirements for the degree of
Doctor of Philosophy
(Electrical Engineering)
at the
UNIVERSITY OF WISCONSIN-MADISON
1985
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
© C o p y rig h t by Robert C harles Becker 1985
A ll R ights Reserved
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
TABLE OF CONTENTS
I.
In tro d u c tio n
1 .1
1.2
1.3
1.4
1.5
II.
H isto ry
In tro d u c tio n to th e concept o f d is tr ib u te d
a m p lific a tio n
A n a ly tic a l expressions fo r gain
Image param eter f i l t e r s
D is trib u te d a m p lifie r gain ex p ressio n s fo r
th e case o f lo s s y tran sm issio n lin e s
1
1*
7
12
16
Gain R elatio n s
20
2 .1
III.
Development o f co n stan t f r a c tio n a l
bandwidth curves
2 .2 S ynthesis o f in d u cto rs u sin g m ic ro s trip
tran sm issio n lin e s
2.3 Using th e co n stan t f r a c tio n a l bandwidth curves
2.1* Design tr a d e - o f f s
2.5 Comparison to a d is tr ib u te d t r a n s i s t o r
28
37
1*5
58
Design C onsiderations
62
3.1
3.2
3.3
3.1*
62
86
95
3.5
IV.
Impedance matching o f image param eter lin e s
S e n s itiv ity a n a ly sis
m -derived tran m issio n lin e s
I n v e s tig a tin g th e v a l id it y range o f th e
alpha ex p ressio n s
A m plifier design u sin g frequency dependent
in d u cto rs
C o n strain t Free Device Design fo r S p e c ific
A p p lications
Device param eter d e f in itio n from a m p lifie r
s p e c if ic a tio n s
1*.2 D iscussion o f p h y sic a l c o n s tr a in ts lim itin g
device param eters
20
110
128
ll*5
1*. 1
ll*5
ll*8
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
V.
A m plifier Power C onsideration
153
5.1
5.2
153
5.3
VI.
Power c a p a b ility o f d is tr ib u te d a m p lifie rs
Comparison o f th e power gain o f
d is tr ib u te d and cascade a m p lifie rs
C o n trib u tio n s o f in d iv id u a l d ev ices in th e
d is tr ib u te d a m p lifie r to o u tp u t power
Conclusion
Appendix I I .
Appendix I I I .
159
168
References
Appendix I .
156
172
O rigin o f th e fa c to r (l-X ^)
gain ex p re ssio n
in th e
17^
Tables o f co n stan t f r a c tio n a l bandwidth
c o e f f ic ie n ts
181*
Comparison o f th e d is tr ib u te d a m p lifie r to
th e d is tr ib u te d t r a n s i s t o r
2 lh
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Constraints in the Design of GaAs MESFET MMIC Distributed Amplifiers
I.
In tro d u c tio n
1 .1
H isto ry
The concept o f d is tr ib u te d a m p lific a tio n was o r ig in a lly proposed
by P e rc iv a l [ l]
o f an
a m p lifie r
improved.
using
as a tech n iq u e whereby th e gain-bandw idth product
co n stru cted
Gain-bandwidth
c o n sta n t-k type
from a
given vacuum tu b e
lim ita tio n s
in te r s ta g e
of
cascade-type
could
be
a m p lifie rs
matching networks were examined
by Wheeler [2] who developed a "bandw idth-frequency index"
(BFl)
which gave th e upper lim it on bandwidth fo r th i s type o f s tr u c tu r e .
G inzton,
et
d is tr ib u te d
and
a l.
[3]
p resen ted
a m p lifie rs .
tr e a te d
only th e
th e
firs t
d e ta ile d
a n a ly s is
of
The a n a ly s is was based upon wave th e o ry
lo s s le s s
case.
A n a ly tic a l ex p ressio n s
fo r
gain f o r co n sta n t-k and m -derived image param eter coupling s tr u c tu re s
were
p re se n te d ,
and s e v e ra l
a lso
tr e a te d .
Subsequently,
c e n te rin g
lin e s
on d is tr ib u te d
followed
[ ^ ,5 ,6 ] .
of
experim ental
and
extended
in v e s tig a te d
fa c to rs
a
fa c e ts
Horton,
a f f e c tin g
lite ra tu re
w ith uniform g rid
and p la te
g rid
in clu d e
a m p lifie r were
of
et a l.
where
to
o f th e
co n sid erab le
a m p lifie rs
a m p lifie rs
th e th e o ry
o th e r
body
[9] rep o rte d th e r e s u lt s
lo ss
th e se
was
not
lo s s e s .
n e g lig ib le ,
Horton a lso
frequency response in th e presence
1
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
2
o f g rid lin e lo s s e s and p resented sim ple c r i t e r i a
frequency resp o n se.
fo r determ ining
B a ssett and K elley [7] and Sarma [8] explored
th e use o f m -derived f i l t e r se c tio n s as a te ch n iq u e fo r c o n tro llin g
th e phase and frequency response o f th e a m p lif ie r .
th e
th e o ry
and
d is tr ib u te d
output
p resen ted
a m p lifie r
tran sm issio n
p a r tic u la r
c irc u it
a
u sin g
li n e s .
g en eral
case
image-matched,
Chen's
o r ie n ta tio n
fo r
ex p ressio n
non-uniform
a n a ly sis
th e
Chen [l6 ] u n ifie d
is
fo r
th e
in p u t
and
lim ite d
am plifying
to
elem ent.
one
The
case o f a r b it r a r y type and o r ie n ta tio n o f th e am plifying element
was tr e a te d by Shapiro [1 5 ].
D is trib u te d a m p lifie rs which make use o f FET's as th e am plifying
elem ents have been dem onstrated [13] and s e v e r a l c i r c u its o p eratin g
w ell
in to
th e
microwave
region
have
a ls o
been
presen ted
[lU,
1 7 ,1 8 ,2 0 ].
The lo g ic a l lim it to th e d is tr ib u te d a m p lifie r is th e
d is tr ib u te d
tra n s is to r.
where th e
and
th e
gate
d ra in
Mclver
m e ta liz a tio n
m e ta liz a tio n
[ll]
proposed
became th e
became th e
a
d is tr ib u te d
FET
in p u t tran sm issio n
lin e ,
o u tp u t tran sm issio n
lin e .
Kopp [12] analyzed t h i s device using coupled mode th e o ry and tr e a te d
th e
e n tir e
tr a n s i s t o r
as
a c tiv e ly
coupled
li n e s .
Podgorski and
Wei [19) expanded th e a n a ly sis and proposed c r i t e r i a fo r optim izing
perform ance.
e x a c tly
lik e
In many r e s p e c ts , th e d is tr ib u te d tr a n s i s t o r behaves
th e
d is tr ib u te d
a m p lifie r.
However,
s ig n if ic a n t
d iffe re n c e s rem ain, and w ill be d iscu ssed in d e t a i l l a t e r in Sec.
II.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
3
Except fo r Horton [17] and Podgorski [19]» l i t t l e
been devoted
to
a m p lifie rs .
Beyer, e t a l .
th e
am plifying
e s ta b lis h in g
element
in
a desig n
procedure
fo r
e f f o r t has
d is tr ib u te d
[21 ] have examined th e GaAs MESFET as
a
d is tr ib u te d
a m p lifie r.
Because o f
p a r a s itic r e s is ta n c e s in th e gate and d ra in o f th e FET, th e a m p lifie r
input and ou tp u t tran sm issio n lin e s a re lo s s y ; The r e s u l t o f th e se
lo sses
has
led
to
th e
p o s tu la te
o f th e
e x ista n c e o f a maximum
bandwidth fo r a d is tr ib u te d a m p lifie r which is a s p e c if ic f r a c tio n
o f th e maximum frequency o f o s c il la t io n o f th e p a r tic u la r MESFET
used.
Recent e f f o r ts
confirm t h i s
p o s tu la te and s p e c if ic d e ta ils
on designing a MESFET d is tr ib u te d a m p lifie r a re provided [2 2 ].
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
h
1 .2
In tro d u c tio n to th e concept o f d is tr ib u te d a m p lific a tio n .
There a re two b a sic ty p es o f microwave a m p lifie rs :
type and th e d is tr ib u te d ty p e .
th e cascade
S everal sta g e s o f a cascade a m p lifie r
are shown in F ig . 1 .2 .1 .
Cell
F ig. 1 .2 .1
Cascade a m p lifie r w ith a r b it r a r y in te r s ta g e matching
netw orks.
As th e name im p lie s, th e s ig n a l cascades from one c e l l to th e n e x t.
The d ra in o f each t r a n s i s t o r
o f th e n ext t r a n s i s t o r ,
l a s t d e v ic e ).
is
co n ju g ately matched to th e
or th e output load
(in th e
A s im ila r conjugate match e x is ts
gate
case o f th e
from th e d riv in g
source to th e g ate o f th e f i r s t tr a n s i s t o r in th e f i r s t c e l l .
The d is tr ib u te d
a
com pletely
a m p lifie r,
d if f e r e n t
shown in F ig .
p r in c ip le .
Here
th e
1 .2 .2 ,
o p erates on
tra n s is to r
c e l l is th e coupling element between two tra n sm issio n lin e s .
in
each
Whereas
th e c e l ls o f th e cascade a m p lifie r a re designed to absorb a l l power
in c id en t a t th e
g a te ,
th e
c e lls
o f a d is tr ib u te d
a m p lifie r a re
designed to absorb minimal amounts o f power in th e g ate c i r c u i t .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
5
The id e a l element fo r use as an am plifying element in a d is tr ib u te d
a m p lifie r
is
a transconductance a m p lifie r:
th a t
is ,
a FET w ith
no in t e r n a l lo s s e s .
CELL
F ig. 1 .2 .2
MESFET d is tr ib u te d a m p lifie r.
To i l l u s t r a t e th e s ig n ific a n c e o f th e d iffe re n c e between cascade
and d is tr ib u te d a m p lifie rs , consider applying a p u lse to th e input
of
each a m p lifie r.
In th e cascade a m p lif ie r , th e p u lse is coupled
to
th e
at
th e d ra in is coupled to th e g ate o f th e next
gate o f th e
f ir s t tra n s is to r,
s ig n a l is passed from d ra in to
a m p lifie d , and th e output
tra n s is to r.
The
g ate between c e lls u n t i l th e l a s t
c e l l where th e s ig n a l a t th e d ra in is coupled to th e load.
In
c o n tr a s t,
d is tr ib u te d
and is
a
pulse ap p lied
to
th e
a m p lifie r propagates down th e
in p u t term in als
g ate tran sm issio n
absorbed in th e gate lin e te rm in a tio n , Rog.
of
a
lin e
As th e pulse
passes through each c e l l , th e t r a n s i s t o r in th e c e l l in je c ts cu rren t
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
6
in to
th e d ra in
li n e ,
along th e d rain lin e .
and e x c ite s waves which begin to
propagate
As th e s e e x cited waves move along th e d rain
li n e , th e forward waves add to th e waves e x c ite d by th e p u lse on
th e
gate
fin a lly
lin e
p assin g
through
subsequent
a re coupled out o f th e a m p lifie r.
c e lls
u n t i l th e waves
The tr a n s i s t o r s
a lso
e x c ite waves in th e backward d ir e c tio n on th e d ra in l i n e ; however,
th e phase r e la tio n s h ip s a re not optimum fo r t h i s e x c ita tio n .
Any
wave so e x cited is absorbed by th e d ra in lin e te rm in a tio n a t th e
in p u t end o f th e a m p lifie r.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
7
1.3
A n a ly tic a l expressions fo r g ain .
The idea o f summing e x cited wave term s in th e d ra in lin e is
th e b asis o f th e s o -c a lle d
"wave th eo ry " which Ginzton
in h is c l a s s i c a l a n a ly sis o f th e d is tr ib u te d a m p lifie r.
[3]
used
Two c e l ls
o f a g e n era lize d a m p lifie r w ith lo s s le s s g ate and d ra in lin e s a re
shown in F ig . 1 .3 .1 .
od
k +1
og
X
X
F ig . 1 .3 .1 .
Two c e l l g en era lize d d is tr ib u te d a m p lifie r.
The
a m p lif ie r s ,
and
transconductance a m p lifie rs .
Aj^+]_
a re
assumed
to
be
id e a l
For th e case o f A]c=Ajc+i =Aq, th e ou tp u t
s ig n a l is given by
Ell. = ElAoK02 (l/6 d + ^ g )
(1 .3 .1 )
2
and
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
8
(1 . 3. 2)
(1L ° ° + 1 L (0R ^ ed))
E3 = Ei A0 Rq2
2
Under th e co n d itio n s 6g=9d=®0>
Eij = EX A0 R02
(1 .3 .3 )
which is th e r e s u l t o f c o n s tru c tiv e in te r f e r e n c e a t p o rt U.
occurs when th e phase v e lo c itie s
eq u al.
Thus, th e
lo s s le s s
This
o f th e g ate and d ra in lin e s
gain
are
fo r one sta g e o f th e two s ta g e ,
N c e ll- p e r - s ta g e a m p lifie r shown in F ig . 1 .3 .2 is
A = Ngjji / R o r r 02
2
(1 .3 .M
where gm is th e d ev ice tran sco n d u ctan ce.
about when th e d ra in
impedance,
impedance Rq i , so t h a t
th e same.
of
were to
Rq2 is transform ed back t o
in p u t and ou tp u t
gate
impedance le v e l
is
For t h i s c a se , i t is p o s sib le to d e fin e an optimum number
devices
a m p lifie r
th e
The term v^Rqi' ^02 comes
which
minimizes
composed
co n sid er
of
th e
cascaded
to ta l
number
d is tr ib u te d
cascading M-stages
of
devices
a m p lif ie r s .
in
If
o f N -devices per s ta g e ,
an
one
th e
o v e r a ll gain would be
Ac = AM = (N gm* /R o rR02)M
I
(1 .3 .5 )
which d escrib e s an a m p lifie r re q u irin g T=N • M d e v ic e s .
Eq. 1 .3 .5 fo r N, s u b s titu tin g th i s
By so lv in g
in to T, d if f e r e n tia t in g T w ith
re s p e c t to M and s e tt in g th e r e s u l t equal to zero , one finds
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
a m p lif ie r .
MESFET d istrib u te d
cascaded
Two stage
Fig. 1 .3 .2 .
R e p r o d u c e d with p e r m i s s io n of th e co p y rig h t o w n er. F u r th e r re p r o d u c tio n proh ibited w ith o u t p e r m is s io n .
10
M = ln|A c |
(1 .3 .6 )
is th e minimum number o f sta g e s re q u ire d to o b tain a gain o f Ac .
In o rd er to
a m p lif ie r ,
fin d th e gain o f a s in g le sta g e o f th e d is tr ib u te d
we s e t M=1 which,
Naperian b a se ).
v ia
Eq.
1 .3 .6 ,
forces
Ac=A=e (th e
The minimum number o f devices per sta g e is then
(1 .3 .7 )
N = 2e___________
Sm / R01R02
One o f th e claims o f th e lo s s le s s th e o ry is th e a b i l i t y
th e
d is tr ib u te d
a m p lifie r
th e
"bandw idth-frequency
to
provide
index"
(BFl)
gain
of
at
th e
freq u en cies
device
of
above
d efined
by
Wheeler [2] as
= gm
1
where
— ;— ~
IT V Ci*0o
Cj_
and
0o
(1 . 3 . 8 )
a re
device
in p u t
and
output
c a p a c ita n c e s,
re s p e c tiv e ly .
It
fmax>
is
im portant to p o in t out th a t
fj
is not th e frequency,
which th e MAG o f th e d evice is u n ity .
The BFI is based
upon th e maximum bandwidth which can be achieved by using c o n sta n t-k
ty p e
in te r s ta g e
[2 ].
fmax is s o le ly a device p ro p e rty , and does not r e f l e c t c i r c u i t
li m ita tio n s ;
it
matching
is
networks
lim ite d
by
to
dev ice
cascade
p a r a s itic
p h y s ic a l d ev ices, f j is always le ss th an fjnax*
th is
id e n tic a l
devices
elem ents.
For
F a ilu re to reco g n ize
leads one to construe f^ as f m x and leads to th e erroneous
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
11
conclusion t h a t th e d is tr ib u te d a m p lifie r can exact gain from devices
o p e ra tin g above f ^ .
This
e rro r
is
commonly found in analyses
based upon a lo s s le s s case a n a ly sis o f th e d is tr ib u te d a m p lifie r.
The presence
of
lo sses
lim its
a v a ila b le
gain
and bandwidth and
r e s u lts in th e appearance o f a maximum gain-bandwidth product equal
to th e gain-bandw idth product o f th e t r a n s i s t o r .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
12
1.1+ Image param eter f i l t e r s .
When
th e
gates
of
th e
FET's
a re
coupled
to g e th e r
v ia
a
s e p a ra tin g in d u c to r, th e r e s u ltin g s tr u c tu r e forms a lumped elem ent
tra n sm issio n
li n e .
The same is
tr u e
fo r th e d ra in s .
This ty p e
o f l i n e , shown in F ig. 1.1*.1, is a c o n sta n t-k type f i l t e r s e c tio n
o f th e g e n e ra l c la ss o f image param eter f i l t e r [23].
T-section
(a)
F ig . 1.1*.1
Tr-section
(b)
C onstant-k f i l t e r s e c tio n s .
The g e n e ra l p ro p e rtie s o f lo s s le s s
in Table l.l+ .l.
For th e lo s s le s s
o f th e two lin e to p o lo g ies
is
is
c a se , only th e image impedance
d if f e r e n t
im portant to note t h i s , as i t
f i l t e r sectio n s a re l i s t e d
( c .f .,
Table l . l * . l ) .
It
i s o f g re a t s ig n ific a n c e t o th e
frequency response o f th e d is tr ib u te d a m p lifie r.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
13
Table 1.1*.1
General c h a r a c te r is tic s o f C onstant-k F i l t e r s .
T -sectio n
it- s e c t ion
( l.l* .la ,b )
Zlir = Ro
\ / l - f 2/ f c2"
(l.l* .2 a ,b )
C utoff frequncy
f c = ___ 1
tt/ l C
( 1 . 1* . 3 )
Phase s h i f t / s e c . 9^ = cos-1 [ l - f 2/ f c2 ] 0* = cos- 1 [ l - f 2/ f c2 ]
(1.1*.!*)
For lo s s le s s co n stan t-k type f i l t e r s , th e phase s h i f t per s e c tio n ,
0,
is
alm ost
lin e a r w ith frequency and is
very n early equal to
The term image impedance d e sc rib e s th e p e r io d ic ity o f impedances
on an
at
image param eter li n e .
s e c tio n
boundaries,
same as th e
The impedance o f th e
and th e
load impedance.
source
This
is
(in p u t)
lin e
re p e a ts
impedance is
g r e a tly d if f e r e n t
th e
from th e
case o f conjugate matching where th e load impedance is th e complex
conju g ate o f th e source impedance.
The problem o f providing an image-match to
n e a tly
solved
w ith another ty p e
m -derived h a lf- s e c tio n .
of
a re a l
image param eter
load
filte r,
is
th e
T -sectio n and ir-se c tio n m-derived f i l t e r s
and t h e i r re s p e c tiv e h a lf-s e c tio n s a re shown in F ig . 1.1*.2.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
S
hT
z0
rnL /Z
H
»U
Ul
u»
• ^
K w
1
M
N
<M
tt
V)
E
Z
o*■»
0
Ul _
<M
s
' w
fc
1
s e c tio n s .
w
>»
O
IV
f-
v>
u —
H
h
u <•*
Ui o
vt —
Uj iv
mL/2
z
0
w»
1
t
IV
£
m-derived
z
ot-H
Fig. l . k . 2 .
ml/2
-j
f ilte r
£
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
15
For
th e
m -derived
h a lf
s e c tio n s ,
th e
norm alized
impedance
a t th e b is e c te d p o rt is given by [23] as
(1.U .5)
fo r th e T - h a lf - s e c tio n , and
( 1. U. 6)
1 - X2(l-m 2 )
fo r th e v - h a lf - s e c tio n .
The key fe a tu re h ere is t h a t , fo r m values
o f about 0 .6 , th e values o f Zpr/g and Zj^/2 a t th e b is e c te d p o rts
a re
n e a rly
c o n sta n t.
Thus, th e m -derived h a lf - s e c tio n
serv es as
a convenient matching c i r c u it between an image param eter lin e and
a re s is tiv e
load
s e c tio n can a lso
The r e l a t i v e
or
source.
A d d itio n a lly ,
th e
m-derived
be used as a lumped element tran sm issio n
m erits
o f m-derived lin e s
in d is tr ib u te d
filte r
lin e .
a m p lifie rs
w ill be tr e a te d in d e t a i l in Sec. 3.3 .
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
16
1.5
D is trib u te d
a m p lifie r
g ain expressions
fo r th e
case
of
lo ssy tran sm issio n lin e s .
The s ig n ific a n c e
fa c to r
in
th e
in v e s tig a te d
o f tran sm issio n
performance o f
by
Horton
lin e
th e
lo sses
d is tr ib u te d
[9 ]. His a n a ly sis
as
a lim itin g
a m p lifie r
only tr e a te d th e
was
case
o f n o n -n eg lig ib le in p u t li n e lo sse s and n eg lected output lin e lo s s e s .
We w ill now examine th e case o f n o n -n e g lig ib le in p u t and ou tp u t
lo sse s which is ty p ic a l o f th e MMIC case.
A ty p ic a l c e l l o f a d is tr ib u te d a m p lifie r is shown in F ig . 1 .5 .1
where Yg and Yd
a re th e g ate
and
d ra in lin e propagation fu n c tio n s .
The v o ltag e gain o f an a m p lifie r c o n stru cted o f uniform in p u t and
outpu t tran sm issio n lin e s may be a s c e rta in e d by wave concepts.
V
d (k-
g(k-1 )
%/2
%n
%/z
Yg
n
V
F ig. 1 .5 .1
d(k)
g(k)
.. .
S in g le c e l l o f a g en era lize d D is trib u te d A m p lifier.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Although th e a m p lifie r could be tr e a te d as a f o u r - p o r t, t h i s would
u n n e c e ssa rily
com plicate
th e
search
fo r a sim ple ex p ressio n
forw ard g ain .
For th e c e l l in F ig . 1.5.1> th e gain is
fo r
(1.5.1)
v dk = Ak v ck EXP(-Yd/ 2 )
or
,
vdk
v
(1. 5. 2)
= Ak * EXE(-(Yd + Yg )/ 2)
I f th e a m p lifie r c o n s is ts
o f N c e l l s , Eq. 1 .5 .2 can be re w ritte n
in term s o f th e in p u t v o lta g e , Vg0 , and th e ou tp u t v o lta g e , Vdn,
as
Vj _
Y 2 - k = Ak EXP(-(Yd+Yg )/2 ) EXP(-(N-k)Yd) • EXP(-kYg)
SO
Armed w ith th e
in th e a m p lifie r,
to th e o u tp u t.
vgo
tr a n s f e r
fu n ctio n o f a s in g le
(1 .5 .3 )
c e l l anywhere
one can now sum th e c o n trib u tio n o f a l l c e lls
I f we assume A^ id e n tic a l fo r a l l k , then
= Ak Exp(-N (,dn g >/2>
B
s in h ( (Yg“Yd ) / 2
<1.5.10
As our in t e r e s t is in GaAs MMIC's, we w ill now examine an a m p lifie r
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
I
18
u sin g MESFET's as th e a c tiv e elem ents and c o n s ta n t-k g ate and d ra in
lin e s .
Wow
Ak = —
2
^ R°g R° d
/ l - f 2/ f cg2 / l + f 2/ f g2
M c o
where gm is th e transconductance o f th e MESFET.
The fa c to r
a r is e s
lin e
from th e tran sfo rm a tio n
o f th e
d ra in
VR0g R0(j
impedance hack
to th e gate lin e impedance so t h a t gain i s measured between p o in ts
at
equal impedances.
The term
g ate v o lta g e on th e
FET w ith
\ / l - ( f / f c g )2 r e f l e c t s
in c re a sin g
frequency.
th e r is in g
This
is
an
in h e re n t p ro p e rty o f a low -loss co n sta n t-k g ate l i n e , and i t s proper
appearance h e re is
deriv ed in Appendix I .
J l + ( f / f g ) 2 c o rre c ts
This
RC c i r c u i t
in c re a sin g
complete
fo r th e s e r ie s
causes
frequency.
v o lta g e
th e
d riv e
The rem aining f a c to r ,
RC in p u t c i r c u i t o f a FET.
to
th e
FET to
S u b s titu tin g Eq. 1 .5 .5
gain
expression
fo r
th e
in to
d ecrease
w ith
Eq. 1.5.**, th e
d is tr ib u te d
a m p lifie r
is
A = Sm
2
\ / Rog Rod
I---------------/ l - f 2/ f cg2
where f cg i s
th e
EXP(-N(Yd+YK)/2 )
/-----------------------/ l + f 2/ f g2
g ate
lin e
,,, , x
(1.5.6)
s i n h ( ( Y g-Yd ) / 2 )
c u to ff frequency,
is th e g ate network frequency.
derived
sinh(N(Yg-y d )/2 )
_1
and f g=( 2irRgCg s )
This same ex p ressio n may a ls o be
u sin g tran sm issio n m a trice s.
This approach was tak en by
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n p rohibited w ith o u t p e r m is s io n .
19
Chen
[U]
in
h is
g en eral case a n a ly s is ,
and r e s u lt s
in th e
same
equation when s p e c ia liz e d to th e case o f id e n tic a l c e l l s .
Q uite c l e a r ly , Eq. 1 .5 .6 hears l i t t l e
s im i la r i ty to Eq. 1 .3 .^ ,
and r e f l e c t s th e r a d ic a l a lte r a t io n o f th e premise upon which th e
a n a ly sis by Ginzton was based.
The previous c r it e r io n fo r optimum
numbers o f devices is meaningless as th e d e sire d gain o f 2.72 per
sta g e o f d is tr ib u te d a m p lifie r may not be p o s s ib le over th e passband
o f th e a m p lif ie r ,
g en era l,
no
tran sm issio n
if at a ll,
and th e a m p lifie r bandwidth i s ,
longer
id e n tic a l
to
th e
c u to ff
li n e s .
I f one assumes th e
frequency
of
phase v e lo c itie s
in
th e
o f th e
gate and d ra in lin e s to be eq u al, i t can e a s ily be shown th a t
N = — - — In
a g- a d
I
ad I
(1 .5 .7 )
gives th e g ain a t any point where otg and a 3 a re known.
There is
no guarantee t h a t th e gain w ill not be le s s th an th e id e a l 2.72
a t some p o in t w ith in th e passband.
A d d itio n a lly ,
performance a r e
th e
not
e f f e c ts
c le a r
of
N,
y,j,
from Eq. 1 .5 .6 .
and
fg
on
a m p lifie r
As w ill be shown in
Sec. 2, f c, fg and Rog a l l in te r a c t through yg , as a re f c and R0(j
coupled through'Y d*
Thus, Rog and R0d can not be s e t a r b i t r a r i l y .
The r o le o f th e s e param eters re q u ire s c l a r i f i c a t i o n .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
20
II.
Gain R elations
2.1
Development o f co n stan t f r a c t io n a l bandwidth cu rv es.
For
purposes
d is tr ib u te d
make th e
on th e
of
a m p lifie r
an aly zin g
which
is
th e
frequency
d escrib ed
response
by Eq.
1 .5 .6 ,
of
th e
we w ill
sim p lify in g assum ption o f ex act phase v e lo c ity matching
gate
and d ra in
li n e s .
A lso,
p e rfe c t
impedance matching
is assumed.
To g en era lize th e a n a ly s is ,
both gain and frequency.
^
we w ill norm alize Eq.
in
Let
_ EXP ("N(ag + ad )/2 ) Sinh(N (ag - ad )/2 )
N / l - X 2 /l+ X 2( f c / f g )2
frequency o f th e a m p lifie r.
(2 .1 .1 )
S in h ((o g - ad )/2 )
be th e norm alized gain o f th e a m p lif ie r .
Sinh(N(ctg - a d )/2 )
1 .5 .6
X =f/fc is th e norm alized
Rote th a t th e fa c to r
(2 .1 .2 )
H S in h ((a g - ad )/2 )
v a rie s s l i g h t l y w ith N, and we w ill examine th i s problem in d e t a i l
s h o r tly .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
21
At
th is
re q u ire d .
p o in t,
a n a ly tic a l
expressions
fo r
ctg and
a re
For th e c o n sta n t-k tra n sm issio n lin e s used h ere,
Cosh y = 1
(2 .1 .3 )
2-Z2
where Z^ and Z2 a re d efin ed in F ig . 2 .1 .1 .
F ig . 2 .1 .1 .
G eneralized T -se c tio n tran sm issio n lin e .
Eq. 2 .1 .3 can be s im p lifie d by expanding th e cosh y fu n ctio n where
Y
= a+jB.
Then
Cosh a Cos 8 = Re
I1 * - ■ - ? ]
( 2 . 1 . Ira)
Sinh a Sin 8 = Im
[ ' -Ar]
( 2 . 1 .Ub)
Using th e id e n tit y cos^ 0 + sin ^ 0 = 1 to elim in a te 8 from E q .'s
2 . 1. 1+a and 2 . 1 .Ub, th e r e s u ltin g equation
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
22
Sinh2 a = Tanh2 a Re2 I" 1 + Zl 1
L
can
be
f u r th e r
s im p lifie d by
sin h 2 a a sta n h 2 a a* a 2 .
Im
+ Im2
2 ' z 2\
assuming
\1
+ ^ L _l
I
(2.1.5)
2 *Z2J
a<0 .k
in
which
case
Under th e s e c o n d itio n s,
1
a » — ------------
- 1/2
(2 .1 .6 )
I1 - Re2 f1 %V]}
is th e g e n e ra l form o f th e a tte n u a tio n c o e f f ic ie n t o f a c o n sta n t-k
tra n sm issio n li n e .
in
F ig .
2 .1 .2 ,
When evaluated fo r th e s p e c if ic networks shown
th e
a n a ly tic a l
form o f th e
g ate and d ra in
lin e
a tte n u a tio n c o e f f ic ie n ts are
_ u2/u ca)g
“g
( l+a)2/u)g2-io2/u)c2) ^/2
ad
~
ojc
=
2~
_2
/kgCgs
(2 .1 .7 a )
2T1/2
(2 .1 .7 b )
=
(2 .1 .7 c )
2
/ Ld^ds
= - 2 -_■■■Rg 'cgs
( 2 .1 .7 d )
= -----1---^ds' ^ds
( 2 .1 .7 e )
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
23
The
s e r ie s
re s is ta n c e
of
th e
in d u cto rs
n eg lecte d to sim p lify th e e x p ressio n s.
has
been
d e lib e r a te ly
The s e r ie s re s is ta n c e and
o th e r com plications w ill be tr e a te d l a t e r .
^
L/2
Lg/2
Ld/2
V 2
ds
'ds
gs
Gate Line
(a)
F ig. 2 .1 .2 .
Drain Line
(b)
Transm ission lin e topology.
E q .'s 2.1.T a & b can be re w ritte n in term s o f th e norm alized
frequency, X, as
° 8 - °°8 X g
(2.1.8a)
«d = ^ L
/ l-X2
(2 .1 .8 b )
“ og = “ c /ug
(2 . 1 . 8c)
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
2h
®od —wd/(i)£
( 2 . 1 . 8d)
g = 1 - 0)c2/a)g2
( 2 . 1 . 8e)
Using th e s e
d e f in itio n s , we can re w rite th e norm alized gain
expression as
An = 1 EXP[-N(aog X2/ V 1-gX2 + a od/ V l-X 2 )/2 ]
N
^ 1-X2 7l+X2(o)c /a)g)2
Sinh[N(aogX2/ / l - g X *
-
a od/ / 1-X2
S in h [(a0g X2/ / l - g X 2 - a od/ / 1-X2
)/2 ]
(2 .1 .9 )
)/2 ]
Following H orton's [9] id e a , we d efin e
a = N a og/2
(2 .1 .1 0 a)
b = M a od/2
( 2 . 1 . 10b)
g = l - ( 2a/N )2
( 2 . 1 . 10c)
and re w rite Eq. 2 .1 .9 u sin g E q .'s 2 .1 .1 0 a & b.
_
EXP[-(a X2/ >/ 1-gX2
+ b / / 1-X2
The r e s u l t is
)]
An
>/ 1-X2 /l+ X 2 (2a/N )2
• |
S inh[a X2 / / 1- gx2
-
b / / 1-X2
Sinh[ (a/N )X2/ \ / T - gx2 - (b /N) /y i - x 2
]
(2 .1 .1 1 )
i
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
25
We now choose "a" and "b" and N as our design c o e f f ic ie n ts .
fo r
la rg e N, th e denominator o f th e second h a lf o f Eq.
Then
2 .1 .1 1
approaches i t s sm all argument v alu e fo r most values o f X.
As such,
th i s
and
term
becomes a sy m p to tic a lly
independent
of
N,
th e
dependence on N mentioned in co n ju n ctio n w ith Eq. 2 .1 .6 has been
circum vented.
Eq. 2 .1 .1 1 now gives us th e norm alized gain in term s o f th re e
c o e f f ic ie n ts :
c o e f f ic ie n ts
N,
"a" and "b".
We a re now fre e to
and examine th e r e s u l t .
F ig .
s e le c t th e se
2 .1 .3 shows th e r e s u l t
o f search in g th e range o f a & b c o e f f ic ie n ts which y ie ld a co n stan t
-3dB bandwidth.
These curves were generated by ign o rin g th e a<0.H
lim ita tio n , and th o se segments o f th e constant-X _3,jB curves which
a re dashed re p re se n t v io la tio n s o f th a t li m it .
is
u s u a lly to o s tr in g e n t,
The oKO.U lim ita tio n
and th e a expressions deriv ed in t h i s
way a re v a lid fo r values above O.U .
Because o f th e s lig h t dependence o f th e norm alized gain eq u atio n
on N ( c . f . :
Eq.
s l i g h t l y w ith N.
2 .1 .2 ) ,
th e
co n stan t-X -j^g curves tend
to
move
For N-values g re a te r th an ^ th e curves a re alm ost
t o t a l l y s ta tio n a r y . Appendix I I c o n tain s ta b le s o f a and b v alu es
fo r N-values from 2 to 10.
The column f-id B /f -3dB g ives a rough
measure o f how f l a t th e response o f th e a m p lifie r i s .
The param eters
o f th e t r a n s i s t o r we choose to u se in our a m p lifie r d efin e an o th er
set
of
curves
on th e
same s e t
o f co o rd in ate ax es.
The curves
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
26
/Div
0.0
Fig. 2 .1 .3 .
1 .4
Constant-X_3,jB ( f r a c tio n a l bandvidth) cu rv es.
in d ic a te p o rtio n s o f curves where accuracy
g u aran teed .
Dashes
is not
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n p rohibited w ith o u t p e r m is s io n .
27
g ra p h ic a lly
be
show tr a d e - o f f s
converted
to
a c tu a l
g a in , & number o f d e v ic e s.
in
norm alized
bandw idth),
bandwidth
impedance,
c u to ff
(which
can
frequency,
In S ect. 2 .h , th e gain-bandw idth product
o f th e a m p lifie r can be ev alu ated on th e same graph by u sin g an
o v erlay .
Thus, th e
e n t ir e a m p lifie r can be designed u sin g th e se
sim ple graphs.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
28
2 .2
S y n th esis o f in d u cto rs u sin g m ic ro s trip tran sm issio n li n e s .
Up to
now, we have assumed th e
d is tr ib u te d a m p lifie r to
be
c o n s tru c te d from lumped in d u c to rs .
In th e case o f MMIC's, lumped
(s o le n o id a l)
g e n e ra lly
a m p lifie rs
in d u cto rs
u sin g
dem onstrated
a re
p rin te d
not
s p ira l
[1 7 ,1 8 ,2 0 ].
and
T y p ic a lly ,
lin e a r
p o s s ib le ,
inductors
although
have
been
a s h o rt piece o f m ic ro s trip
tra n s m is sio n li n e is used as a lin e a r in d u c to r.
F ig .
m ic ro s trip
2 .2 .1
shows
a ty p i c a l
in terco n n ectio n s
being
la y o u t
fo r
employed as
q u ite c l e a r t h a t geom etric co n sid e ra tio n s
an MMIC w ith
in d u c to rs .
s h o rt
It
place a lower lim it on
th e le n g th o f th e in d u c to rs .
DRAIN L I N E
> r
FET
“
M
0
FET
GATE L I N E
F ig . 2 .2 .1 .
is
D is trib u te d a m p lifie r layout lim ited by geometry.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
29
I d e a lly ,
independent.
th is
s y n th e tic
in d u cto r
I f a m ic ro s trip tran sm issio n
should
he
frequency
lin e o f len g th , d,
is
modelled as a iT -section, th e Y-matrix is
Y Coth dY
o
-Y,
Sinh dY
-Y
(2.2.1)
Y Coth dY
o
Sinh dY
where
Y0 = Z0 ^
,
Z0
is
th e
m ic ro s trip l i n e , and d= lin e length.
c h a r a c te r is tic
impedance o f
th e
In th e lo s s le s s case,
Y = jB , and th e c i r c u i t can be rep resen ted as in F ig. 2 .2 .2 a .
- j Y esc
fid
5
Model
C /2
s
Z 5 C /2
s
Lossless
Equivalent
(b)
(a)
F ig . 2 .2 .2 .
ir-sectio n tran sm issio n lin e model.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
30
The
s e r ie s
susceptance.
elem ent,
-jY 0cscBd,
re p re s e n ts
an
in d u c tiv e
I f th e tra n sm issio n lin e is e l e c t r i c a l l y sh o rt ( i . e .
B d « l) , then i t can he e a s ily shown th a t th e e q u iv a le n t inductance
is
Ls = ZQ d/vp
(2 .2 .2 )
where Vp is th e phase v e lo c ity o f th e lin e .
For d< X/T, th e e rro r
is le s s than 1555.
Next, th e shunt elem ent, jY0 ta n Bd/2 must he e v alu ate d .
term
is
s h o r t.
a
c a p a c itiv e susceptance when th e
For th e same
li n e
is
This
e le c tric a lly
r e s t r i c t i o n s as we employed p re v io u sly , th e
eq u iv alen t cap acitan ce is
Cs = d/(Z 0 • Vp )
It
is
im portant to
(2 .2 .3 )
n ote th a t t h i s end
c a p a c ity w ill n o t, in
g e n e ra l, he n e g lig ib le compared to Cgg. The
new g a te lin e c i r c u i t
is shown in F ig . 2 .2 .3 .
The presence o f th e ju n c tio n cap a citan ce,
Cjg, re q u ire s a new ex p ressio n fo r ctg.
A lso, we w ill assume th e
m ic ro s trip
th e
has
a
fin ite
re s is ta n c e
a t t r i b u t e R]_g to th e s e m etal lo s s e s .
in
m etal,
and
we w ill
The shunt d i e l e c t r i c
lo sses
have been neg lected as th ey tend to he overshadowed by th e o th e r
lo s s e s .
Frequency dependence in Rj_g has a lso been n eg lecte d .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
31
R
Lg
/2
L /2
L/2
g
g
sfe
R./2
Lg
c.
gs
F ig. 2 .2 .3 .
C onstant-k T -sectio n g ate tra n sm issio n lin e u sin g
m ic ro s trip
inductors
of
n o n -n eg lig ib le
s e r ie s
r e s is ta n c e .
Eq. 2 .1 .6 is our general form a e q u a tio n , and can be a p p lie d
h ere.
A fter some co n sid erab le a lg e b ra ic m anip u latio n ,
ctg = [u)Rig (Cgg + Cjg ) + o^RgCgs (RlgRgCjg + Lg )]
* ”
+ 2
Lg ^cs s +cJs^
LgRg
~ “ ^Rl Cgs ^Rlg ~ “ 2l,gRgc j g )2
(Cgg + Cjg )(Rqg - <*>2LgRg Cjg )
+ k to2 Lg (Cgg + Cjg) - h u^RgC^ ( R ^ - o)2LgRgCjg)
+ 11 “k i
c % Le <c e s + c Jg> - 11 “ ‘‘" g ^
<Ri g - “ V
s c j g » ' 1 /2
(2.2.U)
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
32
We now d efin e th e follow ing term s:
Rk = p-Rc
2 . 2 . 5a)
Rlg = <l‘Rg = P<l'Ro
2 . 2 . 5b)
2 . 2 . 5c)
L? _
’° =
G gs ^ c jg
2 . 2 . 5d)
dig - l/(R g'C gs)
o)c - 2 /
J Lg(CgS
2 . 2 . 5e)
+ Cjg)
“ eg ” 2/ \l Lg'Cgg
2 .2 .5 f)
“ lg = 2/* J Lg*Cjg
2 . 2 . 5g)
1/a = 1 - q(wc2Au)g2 )
2 . 2 . 5h)
G = 1 + q /2 +
F = q “c
2
+
2 “ lg2
u>lg
“c
- i^ L
l 6wg
2
_
k “ lg 2
i 6u)g2
“g
2
2 .2 .5 J )
2 . 2 . 5k)
2 “ lg 2
2 . 2 . 5m)
«0g -
-S sL ■g "=g2
tpq
+ 1) / / T
(2 . 2 . 5n)
"17
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
33
Using th e se d e fin itio n s i t is p o s sib le to w rite Eq. 2 .2 .h as
ae& = “lg +
e*-2
( 2 .2 .6 )
For ty p ic a l d esig n s, ioc < wg and u>c < u>ig .
reasonable
th a n X.
to
n e g le c t
a ll
term s
of
It
is th en q u ite
frequency dependence g re a te r
Comparing Eq. 2 .2 .6 to E q .'s 2 .1 .8 a & b , i t can be seen
t h a t , for
o„ = a lg + a og x2
g
/T = —
Equation
(2 .2 .7 )
& b.
is
(2 .2 .7 )
n e a rly
id e n tic a l to th e
sum o f E q .'s
2 .1 .8 a
For reasons which w ill become c le a r in Sec. 2 .3 , th e term
a lg
"moved" to Eq. 2 .1 .8 b , as th i s p a rt o f a g a c ts in a manner
id e n tic a l
to
a0(j
in
Eq.
2 .1 .8 b ,
and
has
th e
same
frequency
dependence.
I f we tak e a s im ila r approach to th e d ra in l i n e , th e new d rain
lin e a tte n u a tio n expression is
= a ld + a od
d
-
y n
—
c2
"
( 2 . 2 . 8a)
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
3^
a ld
_
q
(2 . 2 . 8b)
“d
“od = —
“c
(2 . 2 . 8c)
X = u)/air
(2 . 2 . 8d)
“c = 2/ \/ Ld^cds + c jd^
( 2 . 2 . 8e)
wd = l / ( ^ d s ‘^ds^
( 2 . 2 . 8f)
q = Rid/Rds
( 2 . 2 . 8g)
and th e new schem atic
fo r a T -sectio n o f d ra in
lin e is
shown in
F ig. 2 .2 .h .
ds
'ds
F ig. 2 .2 .h.
C o nstant-k T -sectio n d ra in tran sm issio n lin e using
m ic ro s trip
inductors
of
n o n -n e g lig ib le s e r ie s
r e s is ta n c e .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
35
Under th e se circum stances, th e c o e f f ic ie n ts , a and b , d efined
in E q .'s 2 .1 .1 0 a and 2 .1 .1 0 b must be re d e fin e d .
Hence
a = N <*og/2
(2 . 2 . 9a)
b = N(a0d + a id +
become
th e
design
° lg ) / 2
(2 .2 .9 b )
c o e f f ic ie n ts
when
s e r ie s
in d u c to r
en d -cap acitan ce o f th e in d u cto r a re n o n -n e g lig ib le .
m e ta liz a tio n
lo ss
c o e f f ic ie n t
ctig
has
been
lo ss
and
The g ate lin e
a t tr ib u t e d
to
th e
b - c o e f f ic ie n t because i t has th e same frequency dependence as a<j.
This
is
only an approxim ation and i s more f u ll y d iscu ssed in th e
next s e c tio n .
I f one wishes to reduce th e valu e o f th e ju n c tio n c a p a c ito r,
C j,
a h ig h er impedance m ic ro s trip lin e is re q u ire d .
This means,
fo r th e same in ductance, a narrow er and s l i g h t l y s h o r te r p h y sic a l
li n e s iz e .
d e n s it ie s .
For d rain in d u c to rs , t h i s would a ls o mean h ig h e r c u rre n t
T y p ically ,
a
c u rre n t
d e n s ity
o f about
1()5 A/cm^ is
used as th e maximum c u rre n t d e n s ity which gold can c a rry b efo re
conductor degradation due to
e le c tro m ig ra tio n o ccu rs.
It
may be
n ece ssary to in crease conductor w idth (and, hence, ju n c tio n cap a city )
to s a t i s f y t h i s c o n s tr a in t.
E f f e c tiv e ly ,
p aram eters:
If
th e
th e in d u cto r
le n g th ,
len g th
design
impedance (o r
c o n s tra in t
is
has
been bound
by
w id th ),
and c u rre n t
d e n s ity .
removed, E q .'s 2 .2 .2
and
th re e
2 .2 .3
a re
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
36
no longer v a lid and must lie rep laced by
Ls =
r
0)
sin(do)/ Vp)
(d< X/k)
(2.2 .10)
(d< X/U)
(2. 2 .11)
= ^ tan(du>/ 2V-T, )
Z0u
P
which in d ic a te th a t th e s e r ie s inductance d ecreases w ith in c re a sin g
frequency,
w hile
th e
frequency.
As Cs i s
shunt cap acity
shunted
in c re a se s
w ith in c re a sin g
by some d evice c a p a c ity ,
th e e f f e c t
o f a changing Cs is m oderated, and th e LC product tends to d ecrease
as frequency in c re a s e s .
upward w ith
frequency,
Thus, th e poles in a g and
w ith th e
r a te
o f th e m ic ro strip l i n e . The r e s u lt is a
tend to move
dependent upon th e
len g th
d ecrease in th e a tte n u a tio n
when compared to a lin e co n stru cted from a fix e d in d u cto r o f th e
same apparent value a t
low frequencies
(d < X /7).
This can be
used to decrease th e high frequency lo sse s o f th e g ate and d ra in
l i n e s , thereby extending a m p lifie r high frequency response.
Design
u sin g
disadvantage
a b ility
of
to
of
frequency
not
choose
any c o n s tra in t
a n a ly s is :
being
dependent
e a s ily
m ic ro s trip
preclu d es
elem ents
a n a l y tic a ll y
impedance and
th e
p o s s i b ili ty
has
obvious
a c c e s s ib le .
le n g th
of
th e
a
The
independently
g en eral
case
only s p e c if ic cases a re p o s sib le .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
37
2.3 Using th e co n stan t f r a c t io n a l bandwidth curves.
In order to use th e constant-X -g^g curves fo r d esig n , we re q u ire
a
on
te ch n iq u e
th e
whereby th e
graph,
along
tra n s is to r
w ith
th e
param eters
a m p lifie r
can
be d isp la y ed
param eters.
By
using
eq u atio n s 2 . 1. 8a , 2 . 1 . 8b , 2 . 1 . 10a , and 2 . 1 . 10b, one fin d s
a -b = £ f
T
(2 .3 .1 )
U)g
a =
b
ojg tod
(2 .3 .2 )
a re e x a c tly th e equations re q u ire d .
I f one examines Eq. 2 .3 .1 , one fin d s th a t t h i s eq u atio n d efin e s
a
s e r ie s
of
hyperbolas
w ith
f o c ii
along
th e
a=b
li n e .
These
hyperbolas d efin e th e v alues o f th e a and b c o e f f ic ie n ts p o s sib le
w ith a s p e c ific H and a given t r a n s i s t o r under th e c o n s tr a in t o f
eq u al
g ate
c h a r a c te r i s t ic
and
d ra in
lin e
c u to ff
freq u en cies uig and
freq u en cies.
The
device
d e fin e th e d ev ice, hence th e
a*b hyperbolas d e sc rib e th e device in th e a-b p la n e.
These w ill
be c a lle d device c u rv e s.
Equation 2 .3 .2
gives th e slo p e o f a lin e
a d evice curve.
This lin e
drawn from th e o rig in
to
any po in t on
(th e o p e ra tin g lin e )
is
independent o f Nand s p e c if ie s th e c u to ff frequency o f th e lin e s
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
38
in th e d is tr ib u te d a m p lifie r in term s o f th e d evice c h a r a c te r is tic
freq u en cies uig andw<}.
Use o f th i s procedure is shown in F ig . 2 .3 .1 .
s e v e ra l d ev ice curves and two o p eratin g l i n e s .
a
dev ice
o p e ra tin g
o rig in .
curve
in te r s e c ts
p o in t,
and an o p eratin g lin e
(A c tu a lly ,
o p e ra tin g p o in t.
any
a
Here a re shown
The p o in t a t which
constant-X _3(jg curve
point
on th e
is
is
drawn to
dev ice
curve
c a lle d
an
it
from th e
is
a
v a lid
We have chosen a p o in t which l i e s on a p lo tte d
constant-X _3(jB curve fo r purposes o f c l a r i t y . )
The
h y p o th e tic a l
device
curve
is
p lo tte d
in
in te r s e c ts th e X_3(Jb=0.8 curve a t two p o in ts fo r N=2.
#1 is
po in t d esig n ated
chosen, and a l l a sp e c ts
F ig.
2 .3 .1
The o p eratin g
o f th e a m p lifie r
a re now fix e d .
The slo p e of th e o p eratin g li n e determ ines wc fo r
th e
fix e s f-3dg a t x-3dB-f c
lin e s
and
•
By u sin g E q -'s
l.b .2 ,
1.U .3, 2 .1 .7 a , and 2 .1 .7 b one finds
Rog = Rg i
a
(2 .3 .3 )
R0d = ^ H(Js
(2 .3 .M
N
as
th e
T y p ic a lly ,
th e
g ate
and d ra in
Rog is
b - c o e f f ic ie n t
d ra in o f th e FET.
equal to
by th e
lin e
in p u t impedances, re s p e c tiv e ly .
R0(j .
a d d itio n
This
re q u ire s
m o d ificatio n
o f padding c a p a c ito rs
to
of
th e
The n et e f f e c t o f th i s is to modify th e device
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
39
.0
N=2
-3dB
N=1
.10
/Div
0.0
F ig.
2 .3 .1
1 .4
Using th e co n stan t-X -j^g curves and device hyperbolas
to determ ine a m p lifie r param eters.
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
1»0
d ra in tim e c o n sta n t which r e s u lts
in a new s e t o f device cu rv es.
The low frequency g ain is
(2 .3 .5 a )
or
Ao - Bm • \/ ^og'^od
(2 .3 .5 b )
^ e ff
2
w here Ne f f *s
e f f e c t i v e number o f d e v ic e s g iv e n i n T a b le 2 . 3 . 1 .
Thus, s e le c tio n o f an o p eratin g p o in t d e fin e s R0g, R0<j 5 f-3dB»
and A0 ; th i s leaves th e d esig n er l i t t l e freedom.
Taking a d if f e r e n t ta c k , suppose A0 is s p e c if ie d .
Eq. 2 .3 .5 a
can be re w ritte n as
A„ =
(1 - e
Ug
yvR ds
—2b \
)
<*>d
or
b = - 1/2 In
1 -
(2.3.6)
8m V Rg’% s
A g /-d
I t is c le a r ly im possible to r e a liz e an a m p lifie r o f
Ao 2. Sm V Rg • ^ds
y .g /.d
o f a , R0g, N, and X-3dB .
•
'^lis P laces bounds on th e s e le c tio n
Note t h a t th ey a r e not independently
s e le c ta b le .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Ul
< 0 « N O 0 l« a a > n N n < D U > 4 '< 0
O
ONlO(9P)fflnO"ONoMfln-
............................. .............................
9.
od cv >o -o 1/1 in «■ «* «f < 1 0 1 0 1«» to
< OAI Mh 9 ^ t OMS n o ( h Ht
(►
■rt^-t^-H-ocvia)<r'rtOO«o^’ -r4oaj
...........................................................
mN>o<u)tft<^’f M n M n n w
1 0 0 > « i e U l U ) e o e u ) N W 7 N N
09
wtpoino^orothrvvwoffl'Otn
...................................................................................
N<o<ou)in«tnnnni<>niN(«
U»
n N 4 f< to O 0 * « U eO 4 < O 'O
tn<*«N«r-H(h'04’ t9<*4ooors'Oin
l< |eH U i«|(iK 0)«f4B ^e««
<O M o M / i n ^ O > O D N ' O ^ T r O N
n n n N W W N ^ 'H ^ ^ 'rtw ^ ^
0
»
u
0
<►
«►
U1
«<-*4|O(h0>(UC9NOO-*4'OCU»>(D
V l h U l H M ' U 9 M n 4 ’N « i 9 > 0
in^^^nnnoiajtuoiQ ioj-H ^
2 .3 .1
an
0
u
>
0
a
v
o
L.
0
Table
^-N' Oegwr^' OOD^-t ucyTODMo
r 0 N C U G 9 * o N * ’ t u a a 0 ' 0 * ' mc v j
4uiiA «f«nnnnniN iiiN N
M
(U
ftlNU»-0Ch^, oO9t ^ t ^O‘ »<^ts»<
[ s *>cuooDr s ' O' *n( v] ' ' - f ' HOO' 0>
« « i n e N U i | p U ) N e n > ‘H 4 n i O
cD^owrotvj' HochooDr^N' ONOin
ocvj^-ts^moui^rvMocvmcvi
o>CDc^-o<oi nin<' rnmMcucvjcu
1 0 0 0 0 0 0 .0 0 0 0 0 0 0 0 0
Z, fl»«Nrt<-UJ*Ot ^ai O>o<r«NW*, in
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
k2
We can w rite f-3dB=x-3dB’*c as
*-3dB = x~3dB
y oig'ui^'a/b
( 2. 3. 7)
2tt
and lose th i s to e s ta b lis h th e rem aining param eters.
Since we have
chosen a s p e c if ic d evice, and gain (A0 ) is s p e c ifie d , f-3dB> through
s p e c if ie s a unique v alu e o f X_3,jb \/~a fo r which f - 3dB
Eq. 2 . 3 . 7 ,
can be met.
Since b is
x-3dB
lim ite d e s s e n tia lly to th o se values on or very near
is
t h i s b - g r id lin e .
cu rv e,
th e
fix e d , th e search fo r s u ita b le values o f
I f th e re q u ire d v alu e f a l l s below th e N=1 d evice
a m p lifie r
cannot
be c o n stru c te d
using th e t r a n s i s t o r
s p e c if ie d .
At
th i s
p o in t,
a
word about
c a p a c ito r, C(js , i s in o rd e r.
s o ; th e
firs t
is
padding
th e
tra n s is to r
d ra in
There a re s e v e ra l b e n e fits to doing
th e decrease in a 0(j which lowers d ra in
lo s s e s .
This a ls o reduces th e b c o e f f ic ie n t which causes th e device curves
to move c lo s e r to th e o rig in which is th e second b e n e fit.
By moving
th e d ev ice cu rv es, new s e ts o f a and b c o e f f ic ie n ts a re in tro d u ced
which gives th e designer some p re v io u sly u n av ailab le freedom.
Another p o in t to note is in th e development o f equations 2. 3. 1
and 2 . 3 . 2 .
th e
b a sic
These were based on a sim ple,
inductance o f th e tra n sm issio n
com plete forms fo r lin e
lo sse s
lo s s le s s
lin e s .
in d u cto r
as
Using th e more
( 2 . 2 . 7a and 2 . 2 . 8a) d ir e c tly does
not r e s u l t in g en eral s e ts o f constant-X -j^B curves.
The a d d itio n a l
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
1*3
lo ss term s p re se n t in Eq . ' s 2 .2 .7 a and 2 .2 .8 a a re in d u cto r lo s s e s .
For th e d ra in lin e ( 2 . 2 . 8 a ) , th e frequency dependence is th e same
fo r
th e in d u c to r lo ss term as
fo r
th e
lo s s e s
due to Rds a lo n e.
By adding t h i s e x tra lo ss to th e device curves fo r a m p lifie rs u sin g
lo s s le s s in d u c to rs , th e curves move o u t, s l i g h t l y , along th e h -a x is
by th e amount o f th e a d d itio n a l term N*a^^/2 and th e new b -v alu e
is
b ' = b + R c*ld /2 .
( 2 . 3 . 8a)
In c o rp o ra tin g th e lo ss due to in d u c to rs in
th e
device curves i s somewhat more d i f f i c u l t .
lin e
in d u c to rs
has
th e
frequency dependence
and n o t th a t o f th e gate lin e .
a - c o e f f ic ie n t.
equation
th e
A lso,
(2 . 2 . 1 ) ,
e x p o n en tial
h yperbolic
s in e
upon
Thus, i t
exam ination
in d u cto r lo s s )
is
Loss due to
o f th e d ra in
gate
lin e ,
cannot be added to th e
of
th e
norm alized
gain
one sees th a t a g and a d a r e adde'd to g e th e r in
term
term s.
and
In
su b tra c te d
g e n e ra l,
g ate in d u c to r lo ss to th e d ra in li n e .
it
th e gate lin e in to
from
th e n ,
one
another
one cannot
in
th e
a ttrib u te
However, i f a^g (th e g ate
is very sm all compared to a od + a^d (d ra in lo s s e s ) ,
p erm issib le to
"move" th e
lo ss
due to th e g ate in d u c to rs
onto th e d ra in term so th a t
b" = b + N old /2 + H alg /2 .
(2 .3 .8 b )
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
hk
The r e s u ltin g
s ig n if ic a n t,
erro r
and
th e
in
th e
h y p erb o lic
ex p o n en tial
term
s in e
in
ra tio
w ill
not
be
( 2 . 1 . 1 ) w ill dominate
th e r e s u l t .
Hence, gate in d u c to r lo sse s can sim ply be "added" to th e d ra in
term (2 .2 .8 a ) to
get (2 .3 .8 b ) when g a te in d u c to r lo sse s a re sm all
compared to th e t o t a l d rain lin e lo s s e s .
curves outward alo n g th e b -a x is .
This s h i f t s th e dev ice
In r e a l i t y ,
we have moved th e
lo ssy p a rts o f th e tran sm issio n lin e s in to th e device d e s c rip tio n
and sid estep p ed
any
lo ss
o f g e n e ra lity
problems asso c ia te d
in clu d in g in d u c to r lo s s e s in th e constant-X _3<jB cu rv es.
w ith
Note t h a t
as g ate lin e in d u c to r m e ta liz a tio n lo s s e s approach th e t o t a l d ra in
lin e
lo s s e s ,
th e
e rro r
caused by a tta c h in g
b -c o e f f ic ie n t becomes in to le r a b le .
th e om ission o f
which a re le s s
g ate
lin e
th e se
lo sse s
to
th e
By th u s re v e rtin g to Eq. 2 .3 .8 a ,
m e ta liz a tio n
th a n th o se caused by Eq.
lo sse s r e s u lts
2 .3 .8 b .
in e r r o r s
In such c a s e s ,
th e design curves give only rough approxim ations to o v e ra ll a m p lifie r
frequency resp o n se.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p ro d u c tio n prohibited w ith o u t p e rm is s io n .
2. b
Design Tradeoffs
In o rd er to
would
id e a lly
p ro p erly e x p lo it th e d is tr ib u te d a m p lif ie r ,
design
fo r
maximum gain-bandw idth p ro d u ct.
one
Beyer,
e t a l . [2] have shown t h a t , fo r b c o e f f ic ie n ts o f b<0.1t, Eq. 2 .3 .5 a
can be re w ritte n as
(2.1t.l)
where
(2 . It.2 )
is
th e maximum frequency o f o s c i l l a t i o n
fo r a MESFET.
This can
be w ritte n in th e form
( 2 . It.3)
^o *-3dB “ ^K*X_3(jB fnax
where K=(a • b)
is
th e
l /2
norm alized
e
is
th e
norm alized g ain , and X_3,jB=f_3(j g / f c
bandwidth.
Constant-K
over th e
constant-X _3,jB curves
2 . It. 1 ),
and
from th e s e
X_3ds)max ** °*25.
th en Eq.
product
it
o f F ig .
can be
curves
2 .1 .3
(as
can be p lo tte d
shown in
F ig .
shown g ra p h ic a lly th a t
(K •
As th e gain o f th e t r a n s i s t o r is u n ity a t f= f TT,a v ,
2. It. 3 d escrib e s an a m p lifie r which has a gain-bandw idth
which
tr a n s i s t o r
d e v ia te s
by th e
from
fa c to r
th e
gain-bandwidth
ltK*X_3,jB.
ltK ^ a B
is
product
of
th e
norm alized
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
U6
n
.20 .25 -30
35
;3
*4 °
m * 2 *u*1
w ith o u t p erm tesio n -
FurtPerreProduot>on p r o h 'd fted
rmiss'on oUde c o p y r i g ^ ^ '
d
R epro1
u
c e d
^ ' t h
p
e
1*7
gain-bandw idth p ro d u ct, and i t
should be c le a r th a t i t
exceed u n ity in a p h y s ic a l a m p lifie r.
o f th e c o e f f ic ie n ts
a and b in
once a p a r tic u la r device i s
can never
The presence o f c o n s titu e n ts
is n o t cause fo r alarm , as
s e le c te d , a and b a re c o n stra in e d by
th e a*b hyperbolas to a p a r tic u la r range o f v a lu e s.
These values
d e fin e th e tr a n s i s t o r and, th e t r a n s i s t o r has a c o n stan t f ^ y .
S ince
th e
constant-K
and
co n stan t-X -jajj
curves
both
e x is t
in term s o f th e a and b c o e f f ic ie n ts , i t i s p o s sib le t o stu d y th e
behavior o f th e p ro d u ct,
can
be
done
as curves on th e same graph.
by i t e r a t i n g
th e
norm alized gain
p a r ti c u la r a and b c o e f f ic ie n t p a ir .
eq u atio n
This
fo r
a
The K -facto r i s a ls o d efin ed
in term s o f a and b ; by d e fin in g a g rid o f a and b c o e f f i c ie n t s ,
it
is
p o s sib le to
g en era te
a m atrix o f K*X_3,jb p ro d u cts.
This
g rid o f values can be p lo tte d in th re e dimensions and th e r e s u ltin g
curves
form th e smooth s u rfa c e shown in F ig . 2.1*.2.
o f K*X_3,jb values can a ls o
be searched by a
computer
The m atrix
fo r values
which a r e n e a re st to s p e c if ic values o f i n t e r e s t , and th e r e s u lt s
p lo tte d .
So doing g en era tes th e curves shown in F ig . 2.1*.3.
maximum K*X_3,jb product
g re a te r th a n
in v a lid a te
is
0 . 255, which
th e maximum p o s sib le value o f 0 .2 5 .
them,
(namely, sin h
fo r th e se curves
as
(X) »
th e approxim ation used
to
The
is
2%
This does not
d e riv e Eq.
2.1*.3
X fo r X<0.1*) d ev iates by about 3% a t X=0.1*.
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
R eProcluCieel
With Permis
sio n
k9
.150
.175
-3dB
/Div
0.4
F ig . 2 .b .3 .
Constant gain-tandw idth curves w ith constant-X _3^B
curves as a re fe re n c e . The K’X-j^g product is
norm alized gain-bandw idth product divided by fo u r.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
50
The
fact
th a t
th e
curves
o f constant-K 'X _3^g form contours
i s s ig n if ic a n t: th e re a re broad re g io n s on th e constant-X _3(jg curves
where a m p lifie rs o f th e same gain-bandw idth product can e x i s t .
is
This
th e key element in e s ta b lis h in g design tr a d e o f f e f f e c ts .
a m p lifie r
where
th e
design
c o e f f i c ie n t s ,
constant-X _3,jB contour w ill have th e
as any o th e r a m p lifie r where th e
a
and
b,
lie
Any
on
a
same gain-bandwidth product
design c o e f f ic ie n ts
lie
on th e
same constant-X _3^B contour.
Using t h i s
adding
g ate
new to o l ,
a d d itio n a l
and
impedances.
d ra in
it
d ev ices,
lin e s ,
or
To i l l u s t r a t e
is
p o s s ib le to weigh th e m erits o f
changing
c u to ff
changing
g a te
freq u en cies
and
d ra in
fo r
lin e
th e
in p u t
th e u se o f th e s e cu rv es, we begin by
c o n sid erin g th e T ektronix [20] d is tr ib u te d a m p lifie r.
By so doing,
th e th e o r e tic a l p re d ic tio n s may be compared to an a c tu a l d ev ice.
S a lie n t tr a n s i s t o r model param eters a r e given in F ig . 2 . k .k .
G
‘ds
'ds
gs
S
F ig . 2.U.1*.
T ektronix t r a n s i s t o r e q u iv alen t c i r c u i t .
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
51
As th e Tek a m p lifie r has 50 ohm in p u t and output im pedances,
we w ill pad th e d ra in o f th e t r a n s i s t o r w ith 0.310 pF so t h a t th e
low frequency g ate-so u rce and d ra in -so u rc e cap acitan ce is id e n tic a l.
This
e s ta b lis h e s
c u to ff
th e
frequency
equal
e s ta b lis h in g
phase
a re
over th e
p lo tte d
50 ohm d ra in
to
v e lo c ity
th e
impedance and s e ts
g ate
c u to ff
m atching.
Next,
th e
frequency,
th e
d evice
d ra in
th e re b y
curves
constant-X _3<jg and constant-K*X_3(jB cu rv es.
The o p e ra tin g lin e is shown fo r R0 = 50 ohm gate and d ra in li n e s .
This i s shown in Fig. 2.1*. 5.
I t is not n ecessary to c o n sid er th e e f f e c t o f th e n o n -n e g lig ib le
lo s s o f th e in d u c to rs.
The d .c . r e s is ta n c e o f th e s p i r a l in d u c to r
used in th e g ate and d ra in lin e s is about 3 ohms.
and (2 .2 .8 b ) we c a lc u la te a^g = 0.030,
From (2.2.5m )
= 0.031, and a0(j = 0.01*6.
We see t h a t i t is in a p p ro p ria te to a t t r i b u t e a^g to th e b - c o e f f ic ie n t
as a^g i s c le a r ly not sm all compared to a^a + a0(j (th e t o t a l d ra in
lo s s p er s e c tio n o f l i n e ) .
As such, th e p red ic ted response w ill
be o f s l i g h t l y g re a te r bandwidth and gain than th e a c tu a l resp o n se.
The device curves fo r N=6 w ith d ra in in d u cto r lo sse s i s shown
in F ig . 2.1*.6.
50,
Three o p e ra tin g lin e s a re a lso shown: one fo r 1*0,
and 60 ohms.
These correspond roughly to c u to ff freq u en cies
o f 2U, 20, and 16 GHz, r e s p e c tiv e ly .
From th e constant-X _3aB cu rv es,
we see t h a t th e 1*0 ohm lin e gives a -3dB a m p lifie r bandwidth o f
about ll*.5GHz; th e 50 ohm lin e
gives an a m p lifie r -3dB bandwidth
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
52
a
t
f =20GHz
-04/Div
F ig . 2 . It.5.
0.4
A nalysis o f th e T ektronix d is tr ib u te d a m p lifie r w ith
lo s s le s s in d u c to rs and 50 ohm gate and d ra in li n e s .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
53
f = 2 4 .4GHz
0.0
F ig. 2 .b .6 .
.04/Div
f =20GH z
0.4
Tektronix d is tr ib u te d a m p lifie r w ith 3 ohm d . c . r .
inductors in th e d ra in l i n e .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
5b
o f l^GHz,
and th e 60 ohm lin e
o f about 13GHz.
gives a -3dB a m p lifie r bandwidth
The a c tu a l bandwidth w ill be lower as g a te lin e
in d u cto r lo ss has not been accounted f o r .
The
60
ohm
gain-bandwidth
more devices
lin e
product,
(N>6).
comes
c lo s e s t
and th a t could
The fa c t th a t
to
giving
th e
maximum
be improved s l i g h t l y w ith
th e number o f dev ices
e n te rs
in to th e gain-bandwidth product a t a l l should n ot be u n s e ttlin g .
In a
cascade type a m p lifie r,
bandwidth
is
lin e a r :
I f one were to
and
th e
double
th e
re la tio n s h ip
th e
gain and th e
between gain
and
bandwidth h a lv e s.
p a r a l le l H t r a n s i s t o r s , th e gain would be G=N*g0
bandwidth would
be
B=B0/N .
The gain-bandwidth
product
G-B = B0 .go is unchanged.
In
a
d is tr ib u te d
a m p lif ie r ,
and bandwidth is n o n -lin e a r.
th e
r e la tio n s h ip
between
As was shown in Eq. 2 .3 .5 b ,
gain
th e low
frequency gain o f a d is tr ib u te d a m p lifie r is d ir e c tly p ro p o rtio n a l
to
Ne f f ,
th e
e ffe c tiv e
number o f devices
given by Table
2 .3 .1 .
R ecallin g E q .'s 2.1.10b and 2 .1 .8 d , we fin d
b =£ ^
2 oic
This im plies th a t we must move d ia g o n a lly across Table 2 .3 .1 , sin c e
as we change N we change b.
th e
value
of
Ne ff= 2.72.
I f we choose, for example, N=3, b=0.1,
By doubling
Ne ff= ^ .9 ^ ; by t r i p l i n g N, Ne ff= 6 .7 7 .
N,
and
co n seq u en tly ,
b,
C learly He f f is not li n e a r ly
r e la te d to N.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
55
S im ila rly ,
f r a c t io n a l
by examining F ig . 2 . It.5,
bandwidth
(and
hence,
n o n - lin e a r ly r e la te d to R.
th e
is
q u ite c le a r th a t
a c tu a l bandwidth)
is
a lso
In th e case o f p a r a l l e l t r a n s i s t o r s ,
number o f devices d iv id ed out o f th e
because o f th e
R.
it
gain-bandw idth product
lin e a r dependence o f both g ain
and bandwidth on
The n o n -lin e a r dependences o f gain and bandwidth on R in a
d is tr i b u te d
a m p lifie r
do
not
d iv id e
out
of
th e
gain-bandw idth,
p ro d u ct, and a dependence on R rem ains.
The e x iste n c e o f th e maximum gain-bandw idth product as a s in g le
p o in t in s te a d o f a locus i s
a d ir e c t consequence o f th e angular
in c lin a tio n between th e constant-K curves
constant-X _3(jB
h ig h lig h ts
a m p lif ie r :
a
curves
(co n stan t
p rev io u sly
(c o n sta n t gain) and th e
bandwidth)
unknown
p ro p erty
in
F ig .
of
2 . h . i , and
th e d is tr ib u te d
a unique s e t o f c i r c u i t param eters r e s u lt s
gain-bandw idth p roduct.
It
at
a m p lifie r b e s t u t i l i z e d
which a
d is tr ib u te d
merely s ta t e s
th a t th is
th e
in maximum
is th e p o in t
c a p a b ilitie s
of i t s tra n s is to rs .
R eturning to our design example, we see th a t th e 50 ohm lin e
is
s u f f i c i e n t l y clo se
we w i l l
to
not seek to
th e
maximum gain-bandw idth
improve upon i t .
curves shown in F ig . 2.U .7.
p o in t
th a t
There a re th r e e response
The resp o n se fo r 3 ohm d .c . r e s is ta n c e
in th e d ra in in d u cto rs (a=0.90, b = 0.17), fo r 3 ohm d .c . r e s is ta n c e
in
both
resp o n se.
g ate
The
and
d ra in
d e v ia tio n
in d u c to rs ,
of
th e
and
th e
a m p lifie r
a m p lifie r
measured
response w ith
3 ohm
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
56
Gain(dB)
8
♦>H
Q
\
CN
2
2.0
Freq
1 .6GHz/Div
18.0
a > RLg-RLd-0' '
b) % - KU * 3“
c ) Measured
F ig . 2.1+.T.
P red icted v s. measured response fo r th e T ektronix
d is tr ib u te d a m p lifie r.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
57
inductor r e s is ta n c e compared to th e measured response is most lik e ly
due to
d iffe re n c e s
tr a n s is to r s
in
th e
between th e
model t r a n s i s t o r
a m p lif ie r .
The s im ila r ity
response compared to th e measured response is
te s tifie s
to
th e
v a lid ity
and th e
o f th e
a c tu a l
p re d ic te d
q u ite s t r i k i n g and
o f th e technique used to
p r e d ic t th e
frequency resp o n se.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
58
2.5
Comparison
of
th e
d is tr ib u te d
a m p lifie r
to
th e
d is tr ib u te d
tr a n s is to r
The d is tr ib u te d tr a n s i s t o r as proposed by Mclver [ 11 ] is th e
lo g ic a l ex ten sio n o f th e d is tr ib u te d
s tr u c t u r e .
li n e
a m p lifie r in to a continuous
B a sic a lly , th e g ate s t r i p e becomes th e input tran sm issio n
and th e
d rain co n tact
is
th e
output tran sm issio n
lin e ,
as
is shown in F ig . 2 .5 .1 .
d
-WW-
I
OUT
IN
rrr
H -
F ig. 2 .5 .1
For
th is
type
of
D is trib u te d tr a n s i s t o r .
s tr u c tu r e ,
th e
gain
ex p ressio n
(measured
between p o in ts o f equal impedance) is given as [19]
J z6'za
2
<e
- a ~ rlw>
(2 .5 .1 )
’' l ‘ *2
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
59
(2 .5 .2 )
Snnn “ Sn/ \ / 1 + (ui/wg^)
where u>g = l/(R-i/Cg s ) ,
Yl = g a te propagation c o n s ta n t, and 79 =
d ra in propagation c o n sta n t.
This eq u atio n is very s im ila r to Eq.
1 .5 .6 w ith one im portant ex cep tio n .
In S ect. 1 .5 , th e f a c to r
which d efin e s th e lo s s le s s gain o f a c e l l in a d is tr ib u te d a m p lifie r
c o n ta in s
th e
fa c to r
examines
th e
gate
d is c r e t e
tra n s is to rs ,
1-( f / f c g )2
lin e
o f th e
it
can
in
th e
denom inator.
d is tr ib u te d
e a s ily
be
If
one
a m p lifie r which uses
shown,
fo r th e
lo s s le s s
c a s e , t h a t th e vo ltag e a t th e g ate te rm in a ls o f a FET is
(2 .5 .3 )
where Vj i s th e v o ltag e a t th e in p u t te rm in als o f th e g ate l i n e .
E f f e c tiv e ly ,
th e
g ate v o ltag e r is e s
w ith
frequency.
When lo s s e s
a re in tro d u ced , th i s ten d s to compensate them and r e s u lt s in improved
frequency response.
Because th e
com pletely
g ate
d is tr ib u te d
lin e
in
th e
s tr u c tu r e
d is tr ib u te d
as
compared
tra n s is to r
to
a
is
a
d is tr ib u te d
a m p lifie r where th e gate lin e i s co n stru cted from lumped elem en ts,
th e re
is
no
c u to ff
frequency
e s s e n ti a lly i n f i n i t y ) .
drop due to
lo sses
(g ate
lin e
c u to ff
frequency
is
Thus, th e r e is no compensation o f th e v o ltag e
as in th e
lumped li n e .
The absence o f th i s
e f f e c t im plies m onotonically d ecre asin g gain vs. frequency.
This
to
a s s is t
lim ita tio n
th e design
was recognized
o f d is tr ib u te d
and c r i t e r i a
tr a n s is to r s
were in tro d u ced
in th e
p resence
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
60
o f t h i s lim ita tio n .
'o p t
al~ a2
In
The f i r s t a id i s th e optimum le n g th ,
fl
“2
( 2 .5 . U)
This i s id e n tic a l to Eq. 1 .5 .7 except h e re , th e number o f d e v ic e s,
N, i s
rep laced w ith t o t a l g ate w id th , w, due to a change in th e
u n its o f alp h a.
The second a id is a q u a lity f a c to r requirem ent fo r th e g ate
and d ra in li n e s .
P r in c ip a lly , an e x tra p iece o f m etal was connected
to th e g ate as shown in F ig. 2 .5 .2 .
Au
F ig . 2 .5 .2
M odifications o f g ate tra n sm issio n lin e c h a r a c te r is tic s
by a d d itio n a l m e ta lliz a tio n .
This m odifies th e in d u ctan ce, r e s is ta n c e , and cap acitan ce p er u n it
le n g th o f th e g ate li n e .
While t h i s
g re a tly reduces
lo s s e s ,
it
cannot compensate fo r th e monotonic drop in gate v o ltag e as frequency
r i s e s , and a lso
in p u t
impedance reduces g a in .
by Podgorski,
gain
g re a tly reduces
was
th e
* 12dB,
gate
in p u t
and th e
th e
in p u t impedance, and reduced
For th e p a r tic u la r example chosen
impedance was th re e
g ate
s trip e
ohms,
o v e r a ll
was 20000 um long
[1 9 ].
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
6l
C le a rly , some severe p r a c t ic a l problems e x i s t .
F in a lly , i t should be noted th a t th e lumped d ra in lin e o f f e r s
no advantage over a d is tr ib u te d d ra in
lo s s le s s
o u tp u t
lin e .
I f one examines th e
case transim pedance from th e FET d ra in t o th e a m p lifie r
te rm in a ls,
one
fin d s
th a t
th e
magnitude
of
th e
v o lta g e
developed a t th e ou tp u t te rm in a ls o f th e d ra in lin e is independent
o f frequency.
III.
th e
a
These r e la tio n s a r e developed in d e t a i l in Appendix
When lo sses a re in tro d u c e d ,
lo s s e s .
These lo s s e s
ris e
frequency dependence a r is e s
ra p id ly above 80% o f c u to f f
lumped element l i n e ; th e y r i s e
a t a r a te
in
fo r
s im ila r to th e r a t e
f o r a d is tr ib u te d li n e when th e frequency i s below 80!? o f th e c u to ff
frequency o f th e corresponding lumped element lin e .
Thus, a lumped
d ra in c i r c u i t does not o f f e r th e b e n e fits o f a lumped elem ent g ate
c irc u it.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
62
3.
Design Considerations
3.1
Impedance matching o f image param eter lin e s
In th e course o f developing th e gain e q u a tio n s, th e assumptions
o f image matching a t a l l p o rts and between a l l c e l ls was im p lic it.
But r e a l sources and r e a l loads a re not image so u rces and image
te rm in a tio n s .
This p o in t is n ot d iscussed in th e re c e n t d is tr ib u te d
a m p lifie r li t e r a t u r e [1 3 ,1 ^ ,1 7 ,1 8 ,2 0 ], and re q u ire s c l a r i f i c a t i o n .
Consider
a
T -s e c tio n f i l t e r ,
T -sectio n
c o n sta n t-k
Assuming R0 is th e
is
a
lo s s le s s
(3 .1 .1 )
impedance ( r e a l)
o f th e system to which th i s
connected, a t 80# o f c u to f f , r = 0 .2 5 ; a t 90# o f c u to ff,
r = 0 .3 9 .
lo ss
For
th e in p u t impedance is
Z = R0/ >/1 - ( f / f c ) 2
lin e i s
filte r.
These re p re se n t modest mismatch e r r o r s , bu t tran sm issio n
not th e
only c r it e r io n
to
co n sid er. The te rm in a tio n o f
th e g a te lin e can have a s ig n if ic a n t e f f e c t on th e high frequency
respo n se o f an a m p lifie r.
F ig u re 3 .1 .1 shows an example o f th e s ig n ific a n c e o f c o rre c tly
te rm in a tin g th e gate lin e .
The lin e has a 20GHz c u to f f frequency.
At 18GHz, th e v o ltag e a v a ila b le a t th e g ate o f th e l a s t t r a n s i s t o r
is
n e a rly 2dB le ss w ith a r e a l load than w ith th e c o rre c t image
lo ad .
The sig n ific a n c e o f t h i s i s n 't immediately obvious in lig h t
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
R eproduced
with perm ission
of the copyright ow ner. Further reproduction
0]
4J
) \Q
T0D
P
•P in
•H
H
a
i
prohibited
without p e rm issio n .
a) Matched
F ig . 3 .1 .1
26.0
Freq
2.0
2 ‘4 GH z /°iv
b) Unmatched
A comparison o f g a te v o lta g e s a t th e l a s t t r a n s i s t o r in a l |- c e l l d is tr i b u te d
a m p lifie r when r e s i s t i v e and image te rm in a tio n s a re used.
ON
u>
6h
o f th e -3dB drop alre a d y experienced by t h i s t r a n s i s t o r (th e l a s t
in th e image term in ated l i n e ) .
But th e lin e te rm in ated in a r e a l
load i s down 3dB a t 15GHz, n ot 18GHz as is th e image matched li n e .
Thus, th e bandwidth has been in c re ased 20% by u sin g th e c o rre c t
(image) te rm in a tio n .
G eneration
from r e a l
o f approxim ate image param eter
loads and sources
is
re la tiv e ly
sim ple.
in th e d isc u ssio n o f image param eter f i l t e r s ,
th a t th e
m-derived h a lf
c i r c u i t from r e a l to
s e c tio n
loads and sources
In S ect.
1,
i t was pointed out
serv ed as a convenient matching
image lo a d s.
In p a r ti c u la r ,
as F ig.
3 .1 .2
shows, th e match to a r e a l load i s v ery good up to 90% o f c u to ff
f o r iifO .6 in th e low lo ss c a se .
cascade
ty p e
of
a m p lifie r
When compared to a c u rre n t microwave
where
bandwidths
of
1.5
octaves
a re
consid ered very good, t h i s c i r c u i t o f f e r s a trem endously broadband
match.
This
matching c i r c u i t
is
n ece ssary
in th e
g ate te rm in atio n
only when a m p lifie r -3dB bandwidth is TO% o f c u to f f or above.
60% o f c u to f f , th e
mismatch e r r o r
is
Below
n ot s u f f ic ie n t to re q u ire
any matching netw orks, and th e e f f e c t on g ate lin e d riv e a t th e
l a s t c e l l , and hence, a m p lifie r bandw idth, i s n e g lig ib le .
When one considers in p u t and o u tp u t m atching, th e ru le s change
very l i t t l e .
For a ty p ic a l lin e
(g a te o r d r a in ) , th e impedance
Is q u ite co n stan t up to about 10% o f c u to f f .
much b e t t e r t h a t
c u to f f ,
r e tu r n
R eturn lo ss is o fte n
lOdB w ith no m atching s e c tio n s .
lo sses
d ecrease to
Above 10% o f
<10dB, which suggest th e
use
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
of an m-derived
1 /2 -T -s e c tio n .
65
Q)
G
C
CO
“O
CD
a
E
c h a ra c te ris tic s
HH
c
o
•H
4J
G
CD
CO
Normalized
impedance
I
I—
eouepaduix pazxxeuiaoN
Fig. 3.1 .2.
6“
R e p r o d u c e d with p e r m i s s io n of th e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n p rohibited w ith o u t p e r m is s io n .
66
of
matching netw orks.
The
improvement
m -derived h a lf- s e c tio n in th e
in
match w ith
an
m=0.6
g ate lin e i s not s ig n if i c a n t above
80!? o f c u to ff due to th e la rg e in d u c tiv e component o f g a te in p u t
impedance near c u to ff .
The d ra in lin e has le s s in d u c tiv e re a c ta n c e
n e a r c u to f f , and th e match provided by an m-derived h a lf - s e c tio n
i s u s u a lly very good up to 90!? o f c u to ff .
So f a r , we have made few assumptions about th e m atching network
o th e r th an th e value o f m which was s e t a t 0 .6 as p re s c rib e d fo r
lo s s le s s
or very n e a rly lo s s le s s
n e g le c te d .
g a te
and
lin e s .
Any re a c tiv e term s were
In th e presence o f lo s s e s , th e re a c tiv e p a r t o f th e
d ra in
impedances
is
n o n -n e g lig ib le ,
and th e
frequency
dependence o f th e impedance given by (3 .1 .1 ) is no longer a c c u ra te .
Consider th e
b a s ic c i r c u i t s .
c irc u its
shown in
In o rd er to
Lj/2
Lj/2
F ig.
3 .1 .3 .
These a r e th e
o b ta in g en eralized in fo rm atio n about
Lj / 2
Lj /2
Ri
GATE
LINE
(9 )
F ig . 3 .1 .3 .
DRAIN
LINE
<b>
Basic gate and d ra in tran sm issio n lin e c i r c u i t s ,
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
67
th e behavior o f th e s e
c irc u its ,
we begin w ith th e
eq u atio n
for
im age-line input impedance fo r T -sectio n lin e s
Z] _
—
J
where Z]_ is
impedance.
(3.1 .2)
Z]_ • Zp + Z]_2 /I*
th e
s e r ie s
arm impedance and Z2 i s
th e
shunt
arm
Next, we w i l l s c a le th e c i r c u its so th a t R0=l and f c=lHz.
Under th e se c o n d itio n s , Lg=Cg=l/v, R=Rg/R0=p, and f g= l/2 p fo r th e
g a te , and L^=C^=1/tt, R=R(js /Rc =p and f(j=l/2p fo r th e d ra in .
Values
o f p fo r th e g ate ty p i c a ll y range from 0 .1 to 0 .5 w hile p-values
fo r th e d ra in range from 1* to 10.
Using th e n o rm alizatio n s given,
and se p a ra tin g th e impedances in to r e a l and im aginary p a r ts ,
7(1-X 2)2 + l*X2p2 + (1 - X2 )
z ig
1 /2
7 (1 - X2 ) 2 + l*X2p2 - (1 - X2 )
+J
z id
1/2
' 2
’UX2p2
-X2
l*X2p2
/
Jr.
/
\
2
2Xp
1 + l*X2p2
(3 .1 .3 )
1/2
(
\
1 + 1jX2p 2
\
1+X2p2
+J
£
1 + UX2P2
-X2
^x2P2
l*X2p2
1 + UX2p2
1 /2
_ x2
,
(3 .1 .M
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
68
These equations
fo r
a re
shown p lo tte d
th e g ate lin e and F i g . 's
3 .1 .5 a ,b ,
I f th e s e were lo s s le s s T -s e c tio n s ,
in
&
F ig .'s
c for
3 .1 .^ a ,b ,
& c
th e d ra in l i n e .
th e p lo ts would s t a r t a t th e
cen ter o f th e Smith c h a rt
and progress along th e r e a l ax is
toward
zero as
In s te a d ,
h ig h ly
frequency r i s e s .
re a c tiv e a t
p-values as
we fin d th e g ate lin e
low as 0 .2 5 .
behavior is th e d rain lin e .
In sharp c o n tra s t to t h i s
H ere, th e lin e is q u ite r e a c tiv e a t
low fre q u e n c ie s, and, fo r p -v alu es g re a te r than k, c lo s e r to th e
lo s s le s s case as frequency in c re a s e s .
The
to
of
matching s e c tio n s
match lo s s le s s
lo ss
image
introduces
d iscu ssed in
param eter
reac tan ce s
S ect.
li n e s .
in th e
& c and F i g . 's
a re
C le arly ,
gate
n o n -n eg lig ib le and w ill degrade th e match.
examining F i g .'s 3 .1 .^ a .b ,
l.U
designed
th e
lin e
which
comparing them to F i g s .'s
& c where th e
3 .1 .5 a ,b , &c where th e
3 .1 .6 a , b, & c and
p lo tte d
F i g . 's 3 .1 .7 a ,b ,
same impedances a re viewed through an m -derived,
m=0.6, 1 /2 -s e c tio n .
fo r th e
a re
This can be seen by
image impedance functions fo r th e d ra in and gate lin e s a re
and
presence
The term p is th e r a t i o o f Rg/Rc c r Ras /R0
g ate and d ra in
lin e s ,
r e s p e c tiv e ly .
C le a rly , th e d ra in
lin e comes c lo se to th e lo s s le s s case impedance for high freq u en cies
w ith high p -v alu e s.
The g ate lin e has a re a c tiv e component which
in c re ases w ith in creasin g frequency and p -v alu e.
It
is
because o f th e r e a c tiv e term s
fu nctio n s th a t
th e matched lin e s
shown in
in th e image impedance
F ig .'s
3 .1 .6 a ,b ,
& c
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
CURSOR RT X *
.80
,
|S 11|=
.307
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
70
Normalized
Drain
Impedance
.80
|S 1 1|=
0
4.
CURSOR RT X=
,
.266
F ig . 3.1.V b.
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
71
Normalized
Drain
Impedance
0
6.
CURSOR AT X =
.80
,
SI 1
.257
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
72
Normalized
Gate
Impedance
.25
CURSOR RT X™ . 8 0
,
SI 1 = . 2 6 2
F ig . 3 .1 .5 a .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
73
Normalized
Gate
Impedance
oo
p= . 10
CURSOR AT X -
.80
,
I SI 1 -
.252
F ig . 3 .1 .5 b .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Ik
Normalized Gate
p=
Impedance
.05
CURSOR RT X=
.80
,
SI 1
.2 5 0
F ig . 3 .1 .5 c .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
75
Normalized
Drain
Impedance
M atching s e c tio n
m= . 5 0
/
/ \
p*
S . 00
CURSOR AT X=
.8
I
SI 1 I -
. 199
F ig . 3 .1 .6 a .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
76
Normalized
Drain
Impedance
Matohing s e c t io n
m= . 5 0
S '
X
p= 4 . 0 0
CURSOR RT X=
Fig.
.0
9
SI 1 -
.0 99
3 .1 .6 -b .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
77
Normal i z e d
Drain
Impedance
Matching s e c t i o n
m“ . 6 0
S'
p= 6 . 0 0
CURSOR RT X
.8
»
I SI 1 | *
.064
F ig . 3 .1 .6 c .
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
78
Normalized
Gate
Impedance
M a tc h in g s e c t i o n
m= . 6 0
20
CURSOR RT X=
.8
$
SI 1
. 190
Fig. 3.1.7a.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
79
Normalized
Gate
Impedance
M a tc h in g s e c t i o n
m= . 6 0
S
x.
A
Fig. 3.1.7b.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
80
Normalized
Gate
Impedance
M atching s e c tio n
m= . 5 0
S '
V
05
CURSOR RT X =
.8
I
SI 1
.047
F ig . 3 .1 .7 c
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
81
and 3 .1 .7 a ,b ,
se c tio n s
& c degrade so sev erely when m=0.6 m-derived 1 /2 -
a re used as th e matching netw orks.
comes from matching a lo s s le s s
v a rie s as
fu n ctio n
i/l -
.
The value o f m=0.6
lin e where th e impedance fu n ctio n
When th e re a re lo sse s p re s e n t, th e impedance
changes a t a slow er r a te .
The r e s u l t o f th i s
is t h a t ,
a t a given frequency, th e impedance o f a lo s s y T -sectio n lin e is
h igher th a n th e impedance o f a lo s s le s s -line o f th e same c u to ff
frequency and R0 .
The matching se c tio n is e s s e n ti a lly a transform er
w ith a frequency dependent tran sfo rm atio n r a t i o and a t m=0.6, th e
r a t i o is not c o rre c t a t a l l frequencies fo r lo s s y lin e s .
The f a ilu r e
o f th e
th e e n t ir e network.
of
m=0.6 matching network does not condemn
By a d ju s tin g th e m -value, th e r a te o f change
th e tran sfo rm a tio n r a t i o
can he a d ju s te d .
F i g . 's
3 .1 .8 a & b
a re t y p i c a l examples o f th e improvement in match p o ssib le fo r
d ra in
lin e
by changing
from m=0.6
improvements
for th e g ate
fo r m=0.3.
Obviously,
th e
to
m=0.1*.
lin e can be seen
m-value used
in
th e
Equally dram atic
F i g .'s
3 .1 .9 a & b
in th e matching network
can and should be a d ju ste d fo r optimum match.
Thus, by a d ju stin g th e m-value o f th e m atching s e c tio n ,
good m atching can be o b tain ed over th e
e n t ir e
very
bandwidth o f th e
g ate and d ra in tran sm issio n li n e s , and as was shown in F ig. 3 .1 .1 ,
it
is
p o s sib le
to
improve th e
high
frequency
response
of
th e
a m p lifie r by c o r r e c tly te rm in a tin g th e g ate li n e in an image load.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
82
Normalized
Drain
Impedance
Matching s e c tio n
m= . 4 0
co
\
p= 4 . 0 0
CURSOR RT X=
.7
,
SI 1 -
.075
F ig . 3 .1 .8 a .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
83
Normalized
Drain
Impedance
M atching s e c tio n
m= . 4 0
OCl
p= 8 . 0 0
CURSOR RT X=
.8
,
SI 1
.089
F ig . 3 .1 .8 b .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
8k
Normalized
Gate
Impedance
ate
m= . 3 0
P=
CURSOR RT X
.8
9
SI 1
as
.088
Fig. 3.1 .9a.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
85
Normalized
Gate
Impedance
M atching s e c tio n
m= . 3 0
S '
p=
.25
CURSOR RT X=>
.8
I
SI 1
.094
F ig . 3 .1 .9 b .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
86
3 .2
S e n s itiv ity a n a ly s is
In
order to g ain a more p re c is e understanding o f how th e
vario u s
fa c to rs in flu e n c e
th e
performance o f
a m p lif ie r , we tu r n to s e n s i t i v i t y a n a ly s is .
is
an
extrem ely
fa b ric a tio n
u se fu l
to o l
tro u b le s h o o tin g .
fo r
It
a f f e c t perform ance.
d is tr ib u te d
S e n s itiv ity a n a ly sis
a n a ly tic a l work
h ig h lig h ts
th o se
have th e g re a te s t e f f e c t on performance and i t
fa c to rs
a
For s im p lic ity ,
as
w ell
f a c to r s
as
which
shows how th e se
l e t us assume th a t
l ) th e Inductors a re lo s s le s s ; 2) th e r e is no ju n c tio n cap acitan ce
a t th e gate and d ra in co n n ectio n s; 3) 6g = B<j (exact phase v e lo c ity
m atching
in
in d u c to r,
th e passband).
m ic ro s trip
in d u c to r
Except
lo sse s
lo sse s in th e surrounding c i r c u i t .
fo r
th e case
a re
of
g e n era lly
a
s p ir a l
dwarfed
by
By applying stan d ard a n a ly tic a l
tech n iq u es to th e gain eq u atio n ( 1 .5 .6 ) ,
n e
-net,
\
- na
e
- e
f
E
-na
_ c o s h [l/2 (a K- a a ) ]
2 s in h ll /2 /( a - a , ) )
®
8 d
>
>
U)2/u)2g
I
l+u2/u)2-ti)2/a)2
g
eg
(o2/<4
J
( 3 .2 .1 )
1+u2/(o2
g
R
s A as,
n e -n a d
3-n ad _ e-n a g
1
cosh[1 /2 (a g-ad )]
2
s i n h [ l / 2 ( a g-a(3) ]
(3 .3 .2 )
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
87
give the sensitivity of the amplifier, gain, A, to Rg and Rds.
I f we wish to c a lc u la te s e n s i t i v i t i e s f o r th e g ate and d ra in
li n e components, i t is n ece ssary to allo w Bg #
e ffe c t
o f phase v e lo c ity
is in c lu d e d .
mismatch on th e
g ate and d r a in
th e
lin e s
c o s h [ l/2 (y g-Yd ) ]
' nYd _ d~nYg
f
2 s in h [ l/2 ( y g—yd ) ]
/
/
ag <
A
l+(i)2/u 2 - 2u>2/ u)2
eg
g
1 - i2
l+U)2/u>2 —0)2/(l)2
g
Cg
k
<
+ J
f
u)2/U)2
S___
¥ +
l+u)2/(,)2
g
/
th a t
Then, a f t e r some a lg e b ra ,
”n7g
'gs_
so
f
n e
( 3 . 2 . 3)
1 co sh ^1/ g^g~7d^ 1
2 s in h [ l/ 2 ( y g-y d ) ]
-n y g
A
1 + 3u)2/u>2 - 2(d2/u>2
oc _ 1
2
1
V
V.
g
+<02/u>2g -
eg
eg J
aj2/aj2
y
( l + u)2/ uj2 _ ,*,2/^2 ) ^ 2
g
eg
-ny
6
-nyd
)
j
U)/u
2 u )3 / i 0) w*----------------%
eg
i + w2 /w2
(l+oj2/a)2 - o)2/a)2 ) l / 2
eg
y
( 3 . 2 . 1*)
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
eLd _ 1
a “ ¥
^ co sh [1 /2 (yg-y d )]
_
n e
- nYd
>
.
-n y d
2 s i n h [ l / 2 ( Yg-y d )]
e
+ j
¥ +
(3 .2 .5 )
-n y .
n e
-n y .
2 s in h [ l/2 ( y g- y d )]
r
\
(l-w 2/<o2 )1 /2
ca
c o s h [l/2 (y g -y d ) ]
"ds
6
Oj/O)
cd
a d/2
1 “ (i)2/<l)2 ,
cd
-ny
- e
-ny
- e
2u)2/ oj2 - 1 ^
cd
u/w
+ j
1-(D /(I)2
V
cd >
cd
(3.2 .6 )
(l-u>2/a)2d ) 1/2
where
sLg
A
=
s e n s i t i v i t y w . r . t . Lg (g a te li n e in d u c to r)
QCgs
"A
= s e n s i t i v i t y w . r . t Cgs (MESFET g a te ca p a c ita n c e )
S^
A
= s e n s i t i v i t y w . r . t . Ld ( d r a in li n e in d u c to r)
oCds
= s e n s itiv ity
w .r .t.
Cds
(MESFET
d ra in -so u rc e
cap a citan ce)
C le a rly ,
ach iev e
th e
th e s e
hoped
eq u atio n s
fo r
in s ig h t
y ie ld
in to
little
th e
hy
in s p e c tio n .
d is tr i b u te d
To
a m p lifie r
o p e ra tio n , a group o f a m p lifie rs were d esig n ed and th e s e equations
a p p lie d t o them.
The r e s u l t s a re ta b u la te d in Table 3 .2 .1 .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
89
Two groups o f a m p lifie rs were examined.
One design used 150
pm tr a n s i s t o r s : th e o th e r used 300 vm d ev ices.
The g a te li n e and
d ra in lin e impedances were in th e v ic in it y o f 20 ohms and 100 ohms
(re s p e c tiv e ly )
fo r
both
groups
o f a m p lifie rs .
The s ig n ific a n c e
o f th e s e r ie s gate r e s i s t o r in d eterm iin g high frequency perform ance
is
q u ite
c le a r .
T y p ic a lly ,
s e n s itiv ity
to
Rg ranged from about
-1 /1 0 a t 2 GHz to near -1 a t 22 GHz, and in d ic a te s th a t Rg s tro n g ly
in flu e n c e s
high frequency g ain w hile having alm ost no e f f e c t o f
low frequency gain.
In s ta r k c o n tra s t a r e th e
s e n s i t i v i t i e s to R^s which remain
n ear th e same value over most o f th e range from 2 GHz to 22 GHz.
They a r e a lso much la rg e r a t 2 GHz ( ty p ic a lly 0.13 to 0.3M than
a r e th e s e n s i t i v i t i e s to Rg .
This im plies th a t R,js does n o t s tro n g ly
a f f e c t frequency resp o n se, b u t does play a r o le in s e tt in g o v e r a ll
g ain .
Also note th a t th e s e a re p o s itiv e numbers.
in c re a se s
o v e ra ll
g a in ;
in c re a s in g
Rg,
w ith
In c re a s in g Ras
its
n eg ativ e
s e n s i t i t i v i e s , decrease h ig h frequency gain .
The s e n s i t i v i t i e s o f th e gain to L,j and C<js a re more in t e r e s tin g
due to
th e sign changes
in v o lv ed .
Almost a l l o f th e a m p lifie rs
d isp la y ed a change in sig n o f th e s e n s i t i v i t y to one o f th e s e two
components w ithin th e
plus
to
minus
2 to
in d ic a te s
22 GHz passband.
th a t
an
in c re ase
component w ill cause d ecreased g ain .
A sig n change from
in
th e
v alu e
o f th e
The sig n change is
due to
th e changes in gain caused by in c re a sin g lo sses compared to gain
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
90
Rg =11
Rd s =380
Cg s = *44pF
Cds= ‘0 9 1 p F
Lg =.168nH
Ld =.910nH
N=4
f(GHz)
2.000
k.000
6.000
8.000
10.000
12.000
l k . 000
16.000
18.000
20.000
22.000
2 k .000
26.000
28.000
30.000
N=6
f(GHz )
2.000
k.000
6.000
8.000
10.000
12.000
lk .0 0 0
16.000
18.000
20.000
22.000
2 k .000
26.000
28.000
30.000
s g
-.0 1 k
. -.0 5 8
-.1 2 8
~.22k
-.3 k k
-.k 8 k
- .6 k l
-.8 1 3
-.9 9 5
-1 .1 8 5
-1.3T9
-1 .5 7 6
-1.775
-1.975
-2 .1 8 0
R
S g
A
-.0 1 8
-.0 7 3
-.1 6 3
-.2 8 k
-.k 3 3
-.6 0 5
-.7 9 5
-.9 9 8
-1 .2 0 8
- l.k 2 1
-1 .6 3 1
-1 .8 3 6
-2.037
-2 .2 3 6
-2.kk3
ra
sa
A
£A
.2k3
• 2k5
.250
.256
.265
.276
.290
.307
.327
.352
.383
• k21
.k71
.539
.6ki
.128
• 12k
.118
.107
.092
.069
.038
- .0 0 6
-.065
- .lk 7
-.260
- .k l9
-.655
-1.036
-1.762
S Ad
A S
-.0 0 9
-.0 0 9
-.0 1 2
-.0 1 6
-.0 2 3
-.0 3 6
-.0 5 7
-.0 9 0
-.1 3 8
-.2 1 0
-.3 1 7
- .k 8 l
-.7 k k
-1 .2 1 k
-2 .2 2 2
L
s g
A
.252
.2k8
.239
.226
.209
.189
.167
.lk 3
.119
.095
.073
.057
.053
.072
.ik o
L
s ds
A
.3k6
.352
.360
.373
.390
.k l l
.k37
. U69
.507
.553
.609
.678
.765
.883
1.055
£A
.075
.069
.056
.036
.006
-.0 3 9
-.1 0 3
-.1 9 1
-.3 1 3
-.kT9
-.7 0 5
-1.017
-i.k 6 o
-2.135
-3.3k3
c
s gs
A
- . 26k
-.305
-.3 7 2
-.k 6 o
-.5 6 7
-.6 8 6
-.8 1 k
-.9 k 6
-1.076
-1.200
-1.31k
- 1 . kl2 *
-l.k 9 0 *
-1.536 #
-1.532 *
c
SAdS
A
s g
A
s A gs
.095
• 09k
.091
.08k
.071
.0k6
.006
-.057
-.1 5 2
-.2 9 0
-.k 9 2
-.7 8 9
-1.251
-2.0k9
-3 .7 5 1
.257
.256
• 2k9
.238
.226
.215
.207
.20k
.209
.225
.258
.315
.k07
.5kk
.711
-.2 7 0
-.3 2 7
- .k l9
-.5 3 9
-.6 8 1
-.835
-.9 9 5
-1.150
-1.293
- l.k ik
-1.502
-1.5k0
-1.507
-1.370
-1.101
★
Indicates approximations used in the calculations have exceeded
their error b o u n d a r i e s , and the values are therefore s u s p e c t .
Table 3.2.1a
Distributed amplifier sensitivity analysis
for 300um gate width.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
#
*
*
#
91
R =20
g
R. =700
ds
C
gs
=.221pF
L :
g
C. =.046pF
ds
^
N=4
R
f(GHz)
2.000
1+.000
6.000
8.000
10.000
12.000
ll+.OOO
16.000
18.000
20.000
22.000
21+.000
26.000
28.000
30.000
N=8
-.009
-.037
-.082
-.11+1+
-.220
-.309
-.1+08
-.516
-.631
-.750
-.871
-.993
-1.115
-1.235
-1.353
R
R.
S*ds
.132
.133
.131+
.135
.137
.139
.11+1
.11+1+
.11+7
.150
.153
.157
.162
.166
.171
R„
L,
»»d
.181+
.183
.182
.181
.179
.176
.172
.167
.160
.151
.11+0.
.127
.111
.091
.069
L,
f(GHz)
2.000
1+.000
6.000
8.000
10.000
12.000
ll+.OOO
16.000
18.000
20.000
22.000
21+.000
26.000
28.000
30.000
-.013
-.051
-.113
-.196
-.299
-.1+19
-.551
-.692
-.81+0
-.990
-l.ll+O
-1.1+30
-1.1+30
-1.567
-1.697
.252
.251+
.257
.262
.267
.271+
.282
.291
.301
.313
.325
.338
.352
.366
.382
.121+
.122
.120
.115
.109
.099
.081+
.065
.039
.006
-.031+
-.081+
-.11+2
-.2 0 9
-.285
c,
S»ds
-.1 1 8
-.1 1 7
-.1 1 7
-.1 1 7
-.1 1 7
-.1 1 8
-.1 1 9
-.1 2 1
-.1 2 5
-.1 3 0
-.1 3 8
-.11+7
-.1 5 9
-.1 7 3
-.1 9 1
L
s g
A
.21+8
.21+1+
.238
.229
.220
.209
.198
.187
.176
.165
.155
.11+5
.135
.127
.119
C
S gs
A
-.2 5 8
-.2 8 1
-.3 1 8
-.3 6 9
-.1+30
-> 9 9
- .5
-.651+
-.7 3 5
-.8 1 6
-.8 9 6
--97U
-1.01+8
-1.118
-1.181+
c.
l
\ ds
sA
,g
sA
agS
.002
.033
.005
.006
.007
.006
.002
-.0 0 6
-.0 1 9
-.0 3 7
-.0 6 2
-.0 9 5
-.1 3 5
-.1 8 3
-.2 3 8
. 21+7
•239
.228
.217
.207
.199
.195
.197
•203
•216
• 231+
•257
.281+
.315
• 350
-.2 6 3
-.3 0 1
-.3 6 3
-.1+1+1+
-.51+0
-.61+8
-.7 6 1
-.8 7 6
-.9 8 9
-1.097
-1.197
-1.288
-1.369
-1.1+38
-1.1+93
c
Table 3.2.1b
Distributed amplifier sensitivity analysis
for 150um gate width.
R e p r o d u c e d with p e r m i s s io n o f t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n proh ibited w ith o u t p e r m is s io n .
92
in c re ases
due to
exceed gain
improved phase v e lo c ity matching.
in c re a se s
from b e t te r
When lo s s e s
phase v e lo c ity m atching,
sign o f th e s e n s i t i v i t y w ill go n e g a tiv e .
th e
This is more im portant
in long a m p lifie rs (la rg e numbers o f d ev ices) where phase m atching
becomes s i g n if i c a n t,
and can be seen by comparing th e
N=8 150pm d ev ice a m p lifie rs .
S e n s itiv itie s
much
N=8 a m p lifie r
la rg e r
changes
fo r
th e
to
and
L,j and C<js
than
fo r
th e
show
K=U
a m p lifie r.
The behavior o f th e s e n s iti v ity to Lg i s , a t f i r s t appearance,
unusual.
They a re
a ll
p o s itiv e ,
and d ecreasin g w ith
frequency
u n t i l a minimum i s reached, and then in c re a s e w ith frequency.
Much
o f t h i s b eh av io r is due to phase v e lo c ity mismatch, as a minima
l<d
Q
occurs near th e zero cro ssin g in th e corresponding s
o r s Os
A
The f a c t th a t th e minimum is o f f s e t
from th e r i s i n g
e f f e c t is dom inant.
become dominant.
from a zero c ro ssin g comes
gain a sso c ia te d w ith th e r is in g lin e
which accompanies any in crease in Lg.
A
impedance
At low freq u en cies, t h i s
As frequency in c re a s e s , phase v e lo c ity e r r o r s
The minimum occurs a t th e p o in t where impedance
in c re ase induced gain in creases and phase v e lo c ity matching induced
gain in c re ases exchange dominance.
S e n s itiv ity to Cgg is e s s e n tia lly id e n tic a l to th e s e n s i t i v i t y
to
Rg.
Any expected
gain v a ria tio n s
phase v e lo c ity
or
lin e
impedance r e la te d
a re swamped out by th e s tro n g dependence o f th e
gate lin e a tte n u a tio n fu n ctio n on Cgg.
As we have seen in Table
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
93
3 .2 .1 ,
th e
in d iv id u a l s e n s i t i v i t i e s
q u ite h igh a t high fre q u e n c ie s.
o f gain t o
Rg and Cgg a re
The s e n s iti v ity o f g ain t o th e
Rg*Cgs product can not he o b tain ed by th e same te c h n iq u e .
same i s
tr u e
q u a l ita t iv e
fo r th e Fds ’Cds
statem ents
about
pro d u ct.
th e
It
is
s e n s iti v ity
The
p o s sib le t o
by
make
examining
th e
constant-X _3,jB curves (F ig . 2 .1 .3 ) .
We note t h a t , by u sin g Eqs. 2 .1 .8 a , 2 .1 .8 b , 2 .1 .1 0 a and 2 .1 .1 0 b
th a t
a =
Rg'Cgg *(i)c
2 u)g
2
and
b = n oid = fl
1
2 o)c
2 Rds’^ds
1_
“c
By h o ld in g <oc and N c o n s ta n t,
"a"
is
d ir e c tly
p ro p o rtio n a l to
th e Rg’Cgs product and "b" is in v e rs e ly p ro p o rtio n a l to th e Rds’cds
p ro d u ct.
I f ve hold "b" c o n s ta n t, th e spacing o f th e constant-X _3dB
curves as "a" is v a rie d is q u ite c lo se .
bandwidth o f a d is tr ib u te d am p lifer
o f th e Rg’Cgg pro d u ct.
is
This in d ic a te s t h a t th e
indeed a stro n g fu n c tio n
C onversely, holding "a" co n stan t and varying
"b" y ie ld s
a wide sp acin g between co n stan t-x
le t
(Rds'Cds)” 1
f(X)=
-dA/dX
so th a t
th e
^
in v e rs e
X=Rd s -Cd s ,
then
p ro p o rtio n a lity
—3 q B
cu rv es.
I f we
df(X)/dX (dA /df(X))=
o f th e
s e n s itiv ity
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
9^
o f gain to th e Rfls'^ds product m erely r e s u lt s
in a s ig n change
in th e s e n s i t i v i t y , and we can s t a t e th a t th e s e n s i t i v i t y o f gain
t o th e Rds*C<ls product is a r e l a t i v e l y weak fu n ctio n o f th e Rds'^ds
p ro d u ct.
to
th e
I t is c l e a r t h a t th e Rg'Cgg product is very s ig n if ic a n t
high
frequency response
th e Rg'Cgg product is
o f th e
a m p lifie r,
and
reducing
th e way to reduce th e s e n s i t i v i t y to th e
Eg’Cgg product in th e passhand o f th e a m p lifie r.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
95
3.3
M-derived tra n sm issio n lin e s .
In
s e c tio n
3 .1 ,
th e
m-derived
filte r
s e c tio n was d iscu ssed
in co n ju n ctio n w ith th e problem o f p ro v id in g a match between image
param eter loads
and r e a l loads.
was
only th e
v a lid
lim ite d
to
to
co n sid er th e
In t h i s
m-derived
f u ll s e c tio n
c o n te x t, th e d iscu ssio n
h a lf - s e c tio n .
m -derived
It
filte r
is
eq u ally
fo r use
in
th e sy n th e sis o f th e in p u t and o utput tra n sm issio n lin e s tr u c tu re s .
The f u l l s e c tio n m-derived f i l t e r s
These se c tio n s
a re id e n tic a l to th e c o n s ta n t-k f i l t e r s
they a re
d eriv ed except
elem ent.
This
th e
r e s u ltin g
a re shown in Fig.
e x tra
for th e a d d itio n
element
tran sm issio n
c ritic a lly
lin e .
As
3 .3 .1 .
from which
o f one e x tra r e a c tiv e
a lte rs
th e
th e
a n a ly sis
behavior
so
fa r
of
has
concen trated only on th e T -sectio n to p o lo g y , we w ill examine th e
T -sectio n case and comment on th e ir-s e c tio n case.
>L/2
m l/2
mC
T-SECTION
mC/2
^
*m
^ m C /2
tr-SECTION
<fc>
F ig . 3 .3 .1
Lossless m -derived tra n sm issio n lin e s ,
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
96
Consider
th e
T -se c tio n
m-derived
filte r
shown
in
F ig .
3 .3 .1 a .
B a s ic a lly , th e shunt arm o f th e co n sta n t-k f i l t e r has been rep laced
w ith a s e r ie s L-C elem ent.
The impedance o f t h i s branch is
Zs = -JXC ( 1 - (u>/ui0 ) 2 )
(3.3.1)
u>0 = 1 //I T C
The n e t e f f e c t o f th e in d u cto r is to re p la c e th e c a p a c itiv e shunt
element
of
th e
c o n stan t-k
filte r
w ith
a
"frequency
dependent
c a p a c ito r" o f ap p aren t value
C' = C/( 1 - a)2/u>2 )
o
If
th e ca p a c ito r has some
of
th a t
(3 .3 .2 )
shunt lo s s e s ,
th e apparent co n d u ctiv ity
shunt i s s im ila r ly frequency dependent. Forth e
^ -s e c tio n
l i n e , th e s e r ie s arm impedance is
Zs = jX x /d - io2/u>2)
(3 .3 .3 )
which is eq u iv alen t to a frequency dependent in d u cto r o f apparent
value
L' = L /( l - oj2/ o)2 )
o
( 3 .3 d )
Here, any s e r ie s in d u c to r re s is ta n c e a ls o in c re a s e s w ith frequency.
Using th e g e n e ra l form fo r approxim ating a tte n u a tio n
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
97
am
(3 .3 .5 )
IA/C I
>/-UB/C-l
we fin d th e g ate lin e c o e ffic ie n ts to he
(3 .3 .6 a )
A = u3Lg‘Rj*Cgs
B = ( l - u^Ljjjg Cgg)
■C = - “ 2Lg Cgg +
(3 .3 .6 b )
+ <*>2Ri2Cgs2
(3 .3 .6 c )
Lg Cgs
and th e d ra in lin e c o e f f ic ie n ts a re
A —(i)R,js L(j
(3 .3 .7 a )
B = (Rds “ “ 2Rds Lmd cds^2 + “ 2Lind2
(3 .3 .7 b )
C = oj^L(j LmH - u^L(j R(is2 C(js + u)^R(js ^L^
fo r th e lin e s shown in Fig. 3 .3 .2 .
(3 .3 .7 c )
C,js ^
These equations p re d ic t lo sses
which a re g e n e ra lly lower than th e lo sses o f eq u iv alen t c o n stan t-k
tran sm issio n lin e s when m < 1 and examples a re shown in F ig . 3 .3 .3 a
& b.
L9 / 2
Lj /2
Lj / 2
L j/2
Drain
(b)
F ig.
m-derived
c irc u its .
tran sm issio n
lin e s
fo r
g ate
and
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
98
Gate Loss
7
1 .00
dB/Div.
A:
B:
•1 0
Freq.
.40
.0 9 / D i v .
1 .00
a) M :
1.00
b) M ;
.60
g
4
g
F ig . 3 .3 .3 a
Gate
lin e
a) c o n stan t-k
tran sm issio n
so lu tio n s to
th e c i r c u i t in
a tte n u a tio n
fo r
one
T -sectio n
of
tra n sm issio n
lin e
and b) m -derived
lin e .
The curves
re p re se n t
ex ac t
th e a n a l y tic a l ex p ressio n for lo ss o f
F ig . 3 .3 .2 a .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
99
D r a in L o s s
-2
. 10
B:
N:
F ig . 3.3.3b
.40
4
1 . 00
F re q .
09
/D IV
a ) Md:
b)
Md:
1 . 00
. 60
D rain
lin e
a tte n u a tio n
fo r
one
T -se c tio n
of
a) c o n s ta n t-k
tran sm issio n
lin e
and
b) m-derived
tra n sm issio n lin e . The curves re p re s e n t exact so lu tio n s
to th e a n a ly tic a l expressions fo r lo ss o f th e c i r c u it
in F ig . 3 .3 .2 b .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
1 00
Drain
160
Phase
Delay
b
r/
20.00
D sg /D iv
,
^
1
-4 0
. 10
B:
.40
Ni
4
F ig . 3 .3 .^ .
1 . 00
Freq .
09
✓DIV
a) Md:
1.00
b)
.60
Mdi
comparison o f phase delay fo r a) c o n sta n t-k
b) m -derived d ra in tran sm issio n lin e s .
a
and
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
101
The lo s s e s fo r ir-sectio n m -derived tran sm issio n lin e s a re g e n e ra lly
s l i g h t l y h ig h er than f o r a c o n sta n t-k lin e .
The
phase
d if f e r e n t
As
th e
v e lo c ity
th an th a t
m -factor
in c re a s e s .
of
an
o f th e
m-derived
lin e
is
p ro to ty p e co n stan t-k
d e cre ase s,
th e
low
frequency
s ig n if i c a n tl y
filte r
s e c tio n .
phase
v e lo c ity
As th e norm alized frequency, X, approaches c u to f f (X =l),
phase v e lo c ity
for th e
c o n sta n t-k f i l t e r s
m -derived
filte r
remains h ig h er th a n
fo r
o f th e same c u to ff frequency as shown in F ig.
3.3..
A d d itio n a lly ,
th e
m -derived c i r c u its
for th e d ra in and g ate
have a pole in th e t r a n s f e r fu n ctio n s which is not p resen t is th e
c o n sta n t-k
filte r
th e c i r c u i t .
im plem entation.
This
a l te r s
th e
resp o n se
of
The tr a n s f e r fu n ctio n s a re
vgs = ____________________________ ^_____________________
Vi
{[1 + J2P_g Xg - Xg2 ]
[ l- ( l- m 2)Xg2] [1 + kP_g2Xg2 ]}1/2
(3 .3 .8 )
■T =
1
d
/ l - ( l - m 2 )Xd2 + JXd(l-m 2 )/2P_d
Pj* = ^ds/^od
The f i r s t term in th e
th e
frequency
P-£=^ i^ o g
gate v o lta g e tr a n s f e r
dependence
(3 .3 .9 )
of
th e
equation
g a te -to -so u rc e
( 3 .3 .8 )
v o lta g e
c o n sta n t-k f i l t e r s e c tio n ( a t a ir-node) when th e s e r ie s lo ss o f
is
of
a
th e
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
102
network i s
v o lta g e
not assumed to
d iv id e r.
an m -derived
be sm a ll.
The second term
lin e .
When m=l,
The t h i r d
d e p ic ts
both th e
th e
term i s th e R*C
e f f e c ts
o f u sin g
gate and d ra in
tra n s fe r
fu n c tio n s r e v e r t to th e co n sta n t-k form, as th ey should.
The second term in Eq. 3 .3 .8
is
a new p ole in th e tr a n s f e r
fu n c tio n which occurs because o f th e mutual in d u c to r.
S im ila rly ,
a p o le a lso appears in th e t r a n s f e r fu n ctio n o f th e d ra in c i r c u i t
(Eq.
3 .3 .9 ) as a r e s u lt o f th e m utual in d u c to r in th e shunt arm.
This p ole is ab sen t from th e ir-s e c tio n m -derived f i l t e r
is a 2 -p o rt, and th e T -sectio n d ra in lin e i s a 3 -p o rt.
sin c e i t
The presence
o f th e s e poles can be used to modify th e response o f th e d is tr ib u te d
a m p lif ie r .
To complete th e
li n e
image impedances.
gain eq u atio n ,
To o b ta in
d e f in i ti o n o f th e image impedance.
we need th e
th e s e ,
g ate
and d ra in
we w ill r e v e r t to
th e
This eq u atio n is
(3 .3 .1 0 )
Z = Impedance o f s e r ie s element
(3 .3 .1 1 a)
Y = Admittance o f shunt element
(3 .3 .1 1 b )
These expressions can be used to c a lc u la te th e lin e impedance
a t freq u en cies above c u to ff.
The r e s u lt in g norm alized gain equation
is
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
103
A(u) = ~gm / z ig zid____________
2
/ l - ( l - m g2 )Xg2
1
1_______
/l+ U p _ g 2X 2
©
/ 1+j2P.gXg-Xg2
e M(Yg+Y d)/2_______________
Sinh(N (Yg-Yd ) / 2 )
7 l- ( l - m d2 )Xd2+jXd2 (l-m d2 )/2P_d
One a d d itio n a l
m -derived l i n e s .
tim es
as
la rg e
fa c to r
becomes
as
Cd s .
In
freq u en cies o f th e two lin e s
in
lin e
ty p e
(3 .3 .1 2 )
ap p aren t which should
b e n e f it
In a ty p i c a l MESFET, CgS is about th r e e to
o rd e r
v e lo c ity m atching on c o n s ta n t-k
c o n s ta n t-k
S in h ( (yg“Yd ) / 2 )
li n e s ,
im pedances.
th is
to
achieve
approxim ate
g a te and d ra in li n e s , th e
phase
c u to f f
must be approxim ately th e same.
re s u lts
The most u s e f u l
in
CgS.
This
d if f e r e n c e s
co n fig u ra tio n s
employ eq u a l
can be avoided by s e le c tin g
a p p ro p ria te m -value.
be l A
If,
fo r
exam ple,
For
s ig n if ic a n t
impedance lin e s which re q u ire s Cds to be padded to match th e
of
fo u r
d ra in
lin e s
v alu e
w ith th e
CgS=UCd s , th e m-value would
in which case th e d r a in li n e would re q u ire a c a p a c ito r o f
e x a c tly th e v alu e o f Cds to s y n th e s iz e an m=.25 lin e .
The r e s u l t o f such a s e le c tio n process i s , however, e n t i r e l y
unrew arding.
Phase v e lo c ity
erro rs
re s u lt
in
a m p lifie rs
w ith
f r a c t io n a l bandwidths o f X_3dg o f 0 .3 to 0.1+5 w ith no phase v e lo c ity
e r r o r o p tim iz a tio n s and X_3dg in c re a s e s to about 0.5 to 0.65 when
th e d ra in lin e c u to ff frequency i s reduced to improve phase v e lo c ity
m atching.
If
one were to
s u b s titu te
m-derived
filte rs
fo r
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
th e
101+
c o n sta n t-k f i l t e r s
we have "been c o n sid e rin g , th e c u to ff frequency
o f th e g ate lin e would d ecrease.
This would ex acerb ate th e phase
e rro rs a lre a d y encountered, and fo r t h i s reaso n an m -derived gate
lin e has not been given serio u s c o n s id e ra tio n .
A len g th y
d isco u rse onp o ssib le
s o lu tio n s
to
th is
problem
is o f no v a lu e , s in c e in MMIC's, th e in d u c to rs a re fa b ric a te d from
tran sm issio n l i n e s .
The use o f tra n sm issio n lin e s cause a d d itio n a l
cap a c ity to appear a t th e ju n c tio n o f th e s e r ie s and shunt elem ents,
and a t th e t r a n s i s t o r te rm in a ls.
th e c a p a c ity a t th e
When an m -derived lin e is used,
ju n c tio n o f th e s e r ie s and shunt elements is
d if f e r e n t from th e cap a city a t th e t r a n s i s t o r te rm in a ls .
of th is
th e
cap a citan ce was d iscu ssed in s e c tio n 2.2 .
behavior
equations
of
th e
lumped
tra n sm issio n
( 3 .3 .8
and
3 .3 .9 )
no
c h a r a c te r is tic s o f th e lin e s .
longer
li n e s .
a c c u ra te ly
The o rig in
This modifies
The
tr a n s f e r
re fle c t
the
S im ila rly , th e a tte n u a tio n expressions
( 3 .3 .5 , 3 .3 .6 , and 3 .3 .7 ) a re a lso in v a lid .
To t r e a t t h i s
of
a tte n u a tio n
and
case, we must again r e v e r t to th e d e fin itio n s
phase
s h i f t fo r
image
param eter
filte rs .
S p e c if ic a lly ,
Cosh(v) = 1 + ZY/2
(3 .3 .1 3 )
For th e g ate li n e ,
Zg = juLg
(3 .3 .1 ^)
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
105
- W ^iC jm C gg + jttC gg
Yg = > c j g +
(3 .3 .1 5 )
and fo r th e d ra in lin e ,
(3 .3 .1 6 )
Zd = JwLd
l+j(l)RdgCds
(3 .3 .1 7 )
Yd = > c jd +
j^^md “ «)^Ijmd^ds*-'d + ^ds
O bviously, th e se equations a re not r e a d ily e v alu ate d , and a re b est
examined u sin g a computer.
The tr a n s f e r equations fo r th e g ate and d ra in m-derived c i r c u its
a re s im ila r ly com plicated.
l9/z
F ig . 3 .3 .5 .
Lj/2
F ig s. 3 .3 .5 a & b show th e vario u s
L j/2
L j/2
Complete schem atic o f T -se c tio n m-derived tran sm issio n
lin e s for a) th e g ate lin e and b) th e d ra in lin e . Cjg
and Cjd re p re s e n t th e sum o f th e end c a p a c ito rs o f
ir-models o f th e tra n sm issio n lin e s used fo r th e main
inductors and th e mutual in d u c to rs .
For th e d ra in
lin e (b ),
i s th e sum o f th e tr a n s i s t o r C,js and
th e end cap a city o f th e mutual in d u c to r, L^j.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
106
components
and
nodes
in
th e s e
c irc u its .
The
v o lta g e
tra n s fe r
equatio n for th e g ate is
Vc
= [l+fflP(.g + JBg )]
vi
-jk
. T
jo)Lg/ 2
-1
'uL*S
V
(3 .3 .1 8 )
l+jhiR^Cgg
j uCjm_U)^ i^ g s ^ jmg+
1 j j “Lg
^ gs
JL
1+joiR^Cgg
+_
jjw C j j j g - u ^ R i C g s C j j u g + j o i C g s
r
where Yg is defin ed in (3 .3 .1 5 ).
The d ra in c u rre n t v o lta g e tr a n s f e r fu n ctio n is
vd _ h
^
II
id
=
Zds/2
1(i ( jmLd/U + Zid /2 )
ju C jd + _______ 1_________
juLdA + Zid /2
~ U)^iJmd*-'d
JtoLd/U +Zid/2
where Z^d is
th e
+
+
1
“ j “ 3im d ^ jd ^ d “ h>^ iJmdc jd
Rds
Rds
-1
Rd s ( ju>LdA +Zid /2 )
image impedance o f th e
(3 .3 .1 9 )
d ra in
lin e .
The gain
equatio n can now be w r itte n as
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
A(w) =
2
/H s
/!is
(h)
(Z2)
id
vi
V Zid
EXP(-N(Yg+Yd)/2)
Sinh(M(Ys -Y d)/2)
(3 .3 .2 0 )
Sinh((Yg“Yd)/2)
This eq u atio n can he
evaluated fo r v ario u s
lengths and
impedances
o f tran sm issio n l i n e .
This eq u atio n has been compared
to r e s u l t s
published
by
sim u latio n
th e
d a ta .
S chellenberg,
are
norm alized
This
shown
in
equations
response
et
a l.
[2 5 ].
F ig. 3 .3 .6 .
compares
The
fav o rab ly
has been o b tain ed
w ith
not
th e
p u b lish ed
th e
c u to f f
This was done to ach iev e
Note t h a t
our
been changed d e s p ite th e
a d d itio n o f c a p a c ity from th e tra n sm issio n li n e s .
th e sta g g e rin g fa c to r f cd / f cg in t h i s
th e
by stag g erin g
b e t te r phase v e lo c ity matching o f th e two lin e s .
o f c u to ff frequency has
of
The frequency response o f
frequ en cies o f th e two lin e s s l i g h t l y .
d e f in itio n
r e s u lts
Because o f t h i s ,
example is req u ired to b rin g
th e c u to ff freq u en cies o f th e two li n e c lo s e r to th e same v a lu e ,
even though i t appears to push them f u r th e r a p a r t.
One u n fo rtu n a te
asp ect
of
u sin g m -derived lin e
s e c tio n s
is
t h a t th e re is so much la titu d e in d esig n th a t i t is d i f f i c u l t to
choose a s t a r t i n g p o in t fo r a d esig n .
Line length is c o n stra in e d
by th e range o f p r a c tic a l impedances and c u rre n t cap acity ( in th e
case o f th e d ra in lin e ) .
However, co n sid era b le la titu d e is
s till
a v a ila b le .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
108
Response
200
Gain(dB)
10.0/Div
Phase
200/Div
-1800
•
.10
Using Xmission line
A:
B:
N-
1 .02
.33
7
F ig. 3 .3 .6 .
Freq.
1 .00
.09/Div.
1 .00
d'
f cd, / f eg :
.60
1.15
S im ulation o f th e Hughes d is tr ib u te d a m p lifie r [25].
The norm alized frequency X=1 is approxim ately 1*0 GHz,
and th e -3dB bandwidth is 28GHz.
This compares
fav o rab ly w ith th e a c tu a l bandwidth o f 30GHz.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
109
A reaso n ab le approach is to begin w ith a c o n sta n t-k design assuming
equal d ra in and g ate c a p a c itie s ,
th e constant-X _3(jB c u rv e s.
used
to
estim ate
re q u ire d .
th e
th e
From h ere,
tra n sm issio n
and p lo t
The r e s u lt s
frequency
lin e
in d u cto rs
and
device curves
on
from th e se curves can be
resp o n se
one can s e le c t
th e
and
number
o f d evices
le n g th s and impedances
sim u la te
a CAD program to achieve an optim al d e sig n .
th e
r e s u lts
fo r
u sin g
Lines th a t a re very
near to X/h a t th e c u to ff frequency u s u a lly give b est performance
because 'the a tte n u a tio n
is
le ss
a t th e h ig h er freq u en cies.
This
e f f e c t was also d iscussed in Sec. 2 .2 .
Thus,
th e
we have seen th a t th e use o f an m-derived f i l t e r
d ra in
tra n sm issio n
tran sm issio n
lin e
lin e
in d u cto rs
is
a re
of
used.
co n sid era b le
In
a d d itio n
b e n e fit
to
in
when
s lig h tly
lower a tte n u a tio n , th e presence o f a d d itio n a l poles in th e tr a n s f e r
functio n helps to overcome th e e f f e c t o f lo sse s in th e gate c i r c u i t
r e s u ltin g in f l a t t e r a m p lifie r frequency resp o n se.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
110
3.1+ Investigating the accuracy of the ag and
expressions.
In s e c tio n 2.1 we d erived approxim ate a n a ly tic a l expressions
fo r th e a tte n u a tio n p er se c tio n o f th e g a te and d ra in lin e s .
was s ta te d t h a t ,
fo r a <_ 0 . U, th e approxim ations used to d eriv e
th e expressions were v a lid .
th e se ex p ressio n s,
solved .
It
Eq.
In o rd er to
examine th e accuracy of
2 .1 .5 , th e exact eq u atio n fo r a , must be
I f we l e t U=sinh2a and use cosh2a - s in h 2a = l, Eq. 2 .1 .5
becomes q u ad ratic in U, and
a = sin h-1 ( J u ) = ln | /U + (
|u| + l ) 1 / 2 |
(3 .^ .1 )
where
U = - (1 - Im2 [ l + Z1/2Z2 ])
------------ -----------------2(1 - Re2[ l + Z!/2Z2])
(3.!*.2)
^ 1 -In ^ [l+ Z 1/2Z2 ]) + i*(l-Re2 [l+Z1/2Z2 ])(lm 2[l+Z1/2Z2 ])
+
---------------------------------------------------------------------------------------
2(1 - Re2[ l + Z^/2Z2 ] )
d efin es th e d e sire d r o o t.
This
equation
forms th e b asis
o f th e
li m i t , we could
assume
comparison.
$
A d d itio n a lly ,
u sin g
th e
a
_< 0.1*
cosh a to 1 so th a t Eq. 2.1.1*a becomes
B * co s_1[Re[l+Z1/2Z2 ]] .
(3.1*.3)
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Ill
For
T -se c tio n
lin e s ,
we
have,
fo r
th e
g ate
and
d ra in
li n e s ,
re s p e c tiv e ly ,
P g = COS
-1
( 3. b. k )
1-
and
s cos” ^ [ l -2 a)2/o)c2]
(3 .^ .5 )
When a is known, Eq. 2. 1. ka can he solved e x ac tly fo r 6:
B = cos- -*- (R e[l+ Z i/2Z2 l/c o sh a)
The r e s u lt s
o f comparing th e
(3 .^ .6 )
exact expressions
fo r a and B
to th e approxim ate ex p ressio n s a re given in Tables 3 .^ .1 and 3 .^ .2
fo r v arious r a ti o s o f f c/ f g and ^c/^d*
a "typical a p p lic a tio n ,
f c/ f g <_ 0.6
and f c/f d 2.
5. Gate a tte n u a tio n
f c/f g = 1 .0 ,
0 .5 , and
i s shown
re s p e c tiv e ly .
th e
These curves
f c/ f g
up to
< 0 .5 .
3 .^ .la , b,
approxim ate expression
X=0.8 where th e
This
in F ig s.
case o f
and c ,
in d ic a te very good agreement between
exact s o lu tio n and th e
a tte n u a tio n
fo r
0.1
fo r th e
su g g ests
d e v iatio n
th a t
th e
fo r g ate
begins
X=0.9
to
lin e
lin e
in c re ase
on
th e
constant-X _3,jB curves w ill give co n serv ativ e e stim ates o f bandwidth.
Drain lin e a tte n u a tio n
f c/f d r a ti o s
is shown
o f 1, 5, and 11,
in F ig s. 3 .^ .2 a , b,
re s p e c tiv e ly .
& c fo r
Here, th e approxim ate
expression fo r d ra in lo ss d e v ia te s from th e exact s o lu tio n a t both
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
112
Gate Circuit
g:
X
.1
.2
.3
.4
.5
.6
.7
.8
.9
•1
fc/fg :
.2
Exact
.001
.004
.009
.017
.029
.045
.068
.105
.179
fc/fg :
X
.1
.2
.3
.4
.5
.6
.7
.8
.9
Approx
.001
.004
.009
.017
.029
.045
.068
.106
.182
Exact
11.4
22.6
33.4
43.6
53.1
62.0
70.0
77.4
84.1
Approx
.002
.008
.019
.035
.057
.089
.135
.206
.344
Exact
11.4
22.6
33.4
43.6
53.2
62.0
70.2
77.6
84.3
Approx
.003
.012
.028
.052
.085
.132
.197
.297
.474
Exact
11.4
22.6
33.4
43.7
53.3
62.2
70.4
77.9
84.5
Alpha
.3
Exact
.003
.012
.028
.052
.085
.131
.196
.292
.445'
approx
11.4
22.6
33.4
43.6
53.1
61.9
70.0
77.3
84.0
Beta(deg)
Alpha
Exact
.002
.008
.019
.035
.057
.089
.134
.204
.329
X
.1
.2
.3
.4
.5
.6
.7
.8
.9
Beta(deg)
Alpha
Approx
11.4
22.6
33.4
43.6
53.1
61.9
70.0
77.3
84.0
Beta(deg)
Approx
11.4
22.6
33.4
43.6
53.1
61.9
70.0
77.3
84.0
Table 3.4.1
Exact attenuation and phase shift per section of line compared to
approximate values for various ratios of line cutoff frequency to
RC characteristic frequency.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
113
Gate Circuit
Approx
.004
.016
.037
.069
.113
.172
.256
.376
.573
Exact
11.4
22.6
33.5
43.7
53.4
62.4
70.6
78.1
84.7
Beta(deg)
Approx
11.4
22.6
33.4
43.6
53.1
61.9
70.0
77.3
84.0
Approx
.005
.020
.047
.085
.139
.211
.308
.444
.646
Exact
11.4
22.6
33.5
43.8
53.5
62.6
70.9
78.4
84.9
Beta(deg)
Approx
11.4
22.6
33.4
43.6
53.1
61.9
70.0
77.3
84.0
Approx
.006
.024
.056
.101
.164
.246
.355
.500
.700
Exact
11.4
22.7
33.5
43.9
53.7
62.8
71.2
78.7
85.0
Alpha
8x
.1
.2
.3
.4
.5
.6
.7
.8
.9
-4
g,
.5
Alpha
Exact
.005
.020
.046
.085
.138
.209
.303
.430
.598
.1
.2
.3
.4
.5
.6
.7
.8
.9
fc/fe :
gx
.1
.2
.3
.4
.5
.6
.7
.8
.9
Exact
.004
.016
.037
.069
.112
.171
.253
.368
.532
Alpha
.6
Exact
.006
.024
.055
.101
.162
.243
.347
.481
.647
Beta(deg)
Approx
11.4
22.6
33.4
43.6
53.1
61.9
70.0
77.3
84.0
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Gate Circuit
g:
g:
fc /fg :
X
.1
.2
.3
.4
.5
.6
.7
.8
.9
Approx
.008
.032
.073
.132
.210
.309
.432
.584
.770
Exact
11.4
22.7
33.6
44.1
54.0
63.3
71.7
79.1
85.2
Exact
.008
.032
.073
.130
.206
.301
.417
.544
.708
Exact
.009
.036
.081
.144
.226
.326
.444
.579
.727
Approx
11.4
22.6
33.4
43.6
53.1
61.9
70.0
77.3
84.0
Beta(deg)
Alpha
.9
Approx
11.4
22.6
33.4
43.6
53.1
61.9
70.0
77.3
84.0
Beta(deg)
Alpha
.8
X
.1
.2
.3
.4
.5
.6
.7
.8
.9
.064
.117
.187
.279
.396
.546
.740
Exact
11.4
22.7
33.6
44.0
53.9
63.0
71.4
78.9
85.2
Approx
.007
.028
Exact
.007
.028
.064
.116
.185
.274
.385
.522
.683
X
.1
.2
.3
.4
.5
.6
.7
.8
.9
Beta(deg)
Alpha
.7
Approx
.009
.036
.082
.146
.231
.336
.463
.615
.793
Exact
11.4
22.7
33.7
44.2
54.2
63.5
71.9
79.2
85.3
Approx
11.4
22.6
33.4
43.6
53.1
61.9
70.0
77.3
84.0
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
115
Gate Circuit
fc /fg:
X
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
Alpha
Exact
.010
.040
.089
.157
.244
.347
.467
.599
.739
Beta(deg)
Approx
.010
.040
.090
.160
.250
.360
.49.0
.640
.810
Exact
11.4
22.7
33.7
44.3
54.4
63.7
72.1
79.3
85.3
Approx
11.4
22.6
33.4
43.6
53.1
61.9
70.0
77.3
84.0
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
116
Drain Circuit
d‘
X
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.
fc /fd :
3.
fc /fd*
X
.1
.2
.3
.4
.5
.6
.7
.8
.9
Approx
1.005
1.021
1.048
1.091
1.155
1.250
1.400
1.667
2.294
Exact
.434
.599
.721
.826
‘ .926
1.028
1.138
1.258
1.389
Exact
26.6
39.0
49.8
60.0
70.0
79.8
89.3
98.5
106.9
Approx
.335
.340
.349
.364
.385
.417
.467
.556
.765
Exact
17.1
27.8
38.7
50.0
62.1
75.0
89.0
104.2
120.6
Exact
.158
.186
.200
.213
.227
.247
.277
.326
.425
Approx
11.5
23.1
34.9
47.2
60.0
73.7
88.9
106.3
128.3
Beta(deg)
Alpha
5-
Approx
11.5
23.1
34.9
47.2
60.0
73.7
88.9
106.3
128.3
Beta(deg)
Alpha
Exact
.225
.282
.315
.341
.369
.403
.451
.526
.650
X
.1
.2
.3
.4
.5
.6
.7
.8
.9
Beta(deg)
Alpha
Approx
.201
.204
.210
.218
.231
.250
.280
.333
.459
Exact
14.6
' 25.3
36.5
48.3
60.8
74.2
88.9
105.4
124.6
Approx
11.5
23.1
34.9
47.2
60.0
73.7
88.9
106.3
128.3
Table 3.4.2
Exact attenuation and phase shift per section of line compared to
approximate values for various ratios of line cutoff frequency to
RC characteristic frequency.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
117
Drain Circuit
ijj/fj:
7.
.8
.9
fC / f d :
X
.1
.2
.3
.4
.5
.6
.7
.8
.9
Alpha
Exact
.123
.138
.146
.154
.164
.177
.199
.235
.314
X
.1
.2
.3
.4
.5
.6
.7
9.
X
.2
.3
.4
.5
.6
.7
.8
.9
Exact
13.4
24.3
35.8
47.8
60.4
74.0
88.9
105.8
126.2
Approx
.112
.113
.116
.121
.128
.139
.156
.185
.255
Exact
12.8
23.9
35.4
47.5
60.3
73.9
88.9
106.0
127.0
Alpha
Exact
.100
.110
' .115
.120
.128
.138
.155
.184
.248
fC / f d :
.1
Beta(deg)
Approx
.144
.146
.150
.156
.165
.179
.200
.238
.328
Beta(deg)
Alpha
Exact
.084
.091
.094
.099
.105
.113
.127
.151
.205
Approx
11.5
23.1
34.9
47.2
60.0
73.7
88.9
106.3
128.3
Approx
11.5
23.1
34.9
47.2
60.0
73.5
88.9
106.3
128.3
Beta(deg)
Approx
.091
.093
.095
.099
.105
.114
.127
.152
.209
Exact
12.4
23.6
35.3
47.4
60.2
73.8
88.9
106.1
127.4
Approx
11.5
23.1
34.9
47.2
60.0
73.7
88.9
106.3
128.3
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
118
Drain Circuit
d:
Alpha
13.
X
.1
.2
.3
.4
.5
.6
.7
.8
.9
Exact
.073
.077
.080
.084
.089
.096
.108
.128
.174
fc/fd - 15.
X
.1
.2
.3
.4 .
.5
.6
.7
.8
.9
Exact
.064
.067
.069
.073
.077
.083
.093
.111
.151
Beta(deg)
Approx
.077
.079
.081
.084
.089
.096
.108
.128
.176
Exact
12.2
23.5
35.2
47.3
60.1
73.8
88.9
106.1
127.6
Approx
.067
.068
.070
.073
.077
.083
.093
.111
.153
Exact
12.0
23.4
35.1
47.3
60.1
73.8
88.9
106.2
127.8
Alpha
Approx
11.5
23.1
34.9
47.2
60.0
73.7
88.9
106.3
128.3
Beta(deg)
Approx
11.5
23.1
34.9
47.2
60.0
73.7
88.9
106.3
128.3
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
119
Gate Alpha
CTt
,'b
CD Q
a \
Freq.
.09/Div
a) Exact
b) Approximate
f /f =1 .00
c 9
F ig . 3 .^ .1 a
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
120
Gate Alpha
oo
(0
u0)
a
0)
z
1
1 .0
Freq.
.09/Div
f /f
a ) Exact
c
g
= 0.5
b) Approximate
Fig. 3 . l b
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
121
Gate Alpha
in
m
<D in
in
.1
Freq
1 .0
,09/Div
a)
b)
Exact
Approximate
f /f
=0.1
C Q
F ig. 3 .^.1 c
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
12 2
Drain Alpha
Nepi
in
>
•H
Q
o
0.0
a) Exact
1 .0
Freq
.1/Div
b) Approximate
f c^f d = 1
F ig. 3.1*.2a
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
123
Drain Alpha
in
in
Freq
0.0
a ) Exact
1 .0
.1 /Div
f cn/ f H
a = 5
b ) Approximate
F ig . 3.1*.2b
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
12k
Drain Alpha
0.0
a ) Exact
1 .0
Freq
.1 /Div
b) Approximate
fc /fd " 11
F ig. 3.1*.2c
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
125
low (X<0.3) and high (X>0.8) fre q u e n c ie s.
to X=0.8, th e agreement is
X=0.8 ag ain
suggests
th a t
In th e range from X=0.5
reaso n ab ly good.
th e
X=0.9
lin e
The d e v ia tio n above
on th e
constant-X _3<j3
curves w ill give lower bandwidth e stim a te s th an w ill a c tu a lly occur
in a given a m p lifie r.
The low frequency d e v ia tio n is unexpected, but not in e x p lic a b le .
At d . c . , th e propagation c o n s ta n t, y , is undefined: non tim e-v ary in g
fie ld s do not propagate.
Hence, a must go to zero a t zero frequency.
This in fo rm atio n is lo s t in th e approxim ation which shows a f i n i t e
a tte n u a tio n a t d .c .
I t can now be s ta te d w ith c e r ta in ty th a t th e a<0.b lim it is
n e ith e r com pletely c o rre c t nor s u f f ic ie n t to e s ta b lis h e r r o r lim its
in th e a ex p ressio n s.
fo r gate c i r c u i t s
In s te a d , an upper lim it on X o f 0 .8 o r 0.9
o f f c / f g < 0*3 and d ra in c ir c u its
o f f c/ f d
<5
provides b e t te r e rro r c o n tro l.
F in a lly , we note th a t th e p e r-s e c tio n phase s h if ts
fo r g ate
and d ra in lin e s shown in F ig . 3 .^ .3 d e v ia te d ram atically from one
anoth er.
The gate
d ra in lin e
lin e
a n eg ativ e curvature w h ile th e
d isp la y s a p o s itiv e c u rv a tu re .
e n tir e ly
to
an aly sis
which
th e
d if f e r e n t to p o lo g ie s
le d
to
constant-K*X_3,jB curves,
r e s u lt
d isp la y s
o f th is
th e
This v a r ia tio n i s
o f th e two lin e s .
development
In
o f th e constant-X
phase s h i f t was e n tir e ly n e g le c te d .
due
th e
and
The
phase v e lo c ity mismatching w ill cause a drop
in
th e high-X_3,jB curves which w ill b rin g them c lo se r to th e b - a x is .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
126
Phase Shift
o
CNl
ea)no\
Q
CM
0.0
Freq
1 .0
.1 /D iv
a) Gate line:
f /f
c
q
=0.5
b) Drain line: f /f. = 5
c
F ig . 3 .^ .3 .
a
Comparison o f phase v e lo c ity matching between gate
and
d ra in
tra n sm issio n
lin e s
of
eq u al
c u to ff
freq u en cies.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
12 7
As th e amount o f displacem ent v a rie s w ith N, th e number o f d e v ic e s,
it
is n ot p o ssib le to make a g en era l s e t o f c o n s ta n t-X -^ g curves
fo r any case where gate and d ra in phase v e lo c ity a re not e x a c tly
equal.
I t i s , however, p o s sib le to minimize th e phase e r r o r by s h if ti n g
d ra in
lin e
frequency.
c u to ff
frequency upward r e l a t i v e
By so
"stag g erin g "
th e
lin e
to
c u to ff
g ate
lin e
c u to ff
fre q u e n c ie s,
th e
phase v e lo c ity o f th e d ra in lin e i s in creased and th e phase s h i f t
p e r-s e c tio n o f th e lin e is decreased.
This is an e n t ir e ly em p irica l
process and may not g r e a tly b e n e fit performance in sh o rt a m p lifie rs
(sm all H) or a m p lifie rs
to
was
th e
o p eratin g
below X=0.5.
This
is
s im ila r
idea developed by Sarma [ 8 ] where a phase v e lo c ity e rro r
d e lib e r a te ly
a m p lifie r
phase
intro d u ced
lin e a rity
H ere, th e phase v e lo c ity
v ia
and
e rro r
th e
reduce
is
d ra in
gain
c irc u it
to
improve
peaking above X=0.7.
introduced a t midband so th a t
th e high frequency phase v e lo c ity e rro r is reduced, th e re b y improving
th e
h igh
frequency
gain
and
extending
th e
a m p lifie r
frequency
respo n se.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
1 28
3.5
A m p lifier desig n using frequency dependent in d u c to rs.
In th e d is tr ib u te d a m p lifie r co n sid ered so f a r , th e gate and
d ra in
tra n sm issio n
independent
lin e s
components
were
( c h ie f ly ,
assumed
th e
composed
in d u c to rs ).
of
In
frequency
S ection
2,
th e design o f quasi frequency-independent in d u c to rs from m ic ro strip
tra n sm issio n lin e was d iscu ssed .
Here, th e lim it o f A/7 fo r th e
m ic ro s trip le n g th in th e passband o f th e a m p lifie r became th e upper
bound on th e le n g th .
I f th e s h o rt
lin e
length
lim it i s
it
p o s sib le
removed, th en Eqs. 2 .2 .2
and 2 .2 .3 become
Using th e s e
eq u atio n s,
is
to
s e le c t m ic ro strip lin e
le n g th , d, such t h a t th e frequency dependence o f Lg and Cs in th e
passband o f th e am p lifie r is
no longer n e g lig ib le .
This a f f e c ts
th e c h a r a c te r i s t ic s
o f th e d is tr ib u te d a m p lif ie r 's
lin e s in two ways.
F i r s t , th e lin e in p u t impedances, Rog and
w ill d ecre ase w ith
frequency.
frequency
dependence
a sso c ia te d
This
w ith
is
d is tin c t
c o n sta n t-k
gate and d ra in
^
from th e normal
lin e s .
Cs
is
norm ally o f th e o rd er o f CgS ; conseq u en tly , th e frequency dependence
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
129
o f Ls dominates th e in p u t impedance.
dependence,
th e
in p u t
As Ls has a s in ( x ) /x frequency
impedance
of
th e
lin e s
d ecreases
w ith
in c re a s in g frequency.'
One
of
tran sm issio n
o f th e
th e
most
lin e
obvious
in d u cto rs
in p u t impedance.
3 .5 .2 a ,b ,c
fo r
g ate
and
e f f e c ts
of
"long"
is
a
decrease
This
is
shown in F ig s.
d ra in
li n e s ,
in th e
m ic ro s trip
r e a c tiv e
p a rt
3 .5 .1 a ,b ,c
and
re s p e c tiv e ly .
The
fa c to r
p i s th e r a ti o o f p a r a s it ic r e s is ta n c e ( e ith e r Rj or R<js ) to lin e
R0 .
Theta
is
th e
e le c tric a l
in degrees a t f= fc .
le n g th
shows a s ig n if ic a n t
when e l e c tr i c a ll y
in d u c to r
A 5 degree li n e re p re se n ts e s s e n tia lly a pure
in d u c to r w ith no end c ap a city .
g ate
o f th e m ic ro s trip
For th e
re d u c tio n
long lin e s
a re
sample lin e s
shown, th e
in
reactan ce near th e
used.
S im ilar r e s u lts
c u to ff
can not
be ob tain ed for th e d ra in c i r c u i t due to th e to p o lo g ic a l d if f e r e n c e s .
Here, th e reactance i s
la rg e s t a t
low freq u en cies, and th e e x tra
c a p a c ity from th e ends o f th e m ic ro s trip inductors has l i t t l e e f f e c t .
Because of th e a d d itio n a l elem ents and th e ir frequency dependent
beh av io r,
th e a tte n u a tio n
c h a r a c te r is tic s
of th e
a l te r e d .
Losses
gate and d ra in
lin e s
fo r
th e
len g th s a re shown in F ig s. 3.5*3 and 3 .5 .^ .
red u ctio n s in lo sses can be ach iev ed .
or not th i s
lin e s
are
a lso
fo r v ario u s
lin e
C le a rly , s ig n if i c a n t
What is not c le a r is whether
is o f g re a t b e n e fit due to concurrent re d u c tio n s
in
in p u t impedance.
In order to b e t te r understand th e p o ssib le b e n e f its , a q u a lity
fa c to r is needed.
As in p u t impedance and a tte n u a tio n both a f f e c t
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
130
Normalized
T h e t a=
Gate
Impedance
5.00
oo
p=
.20
CURSOR AT X=
.8
F ig . 3 .5 .1 a
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
131
Norma]ized
Theta=
Gate
Impedance
45.00
20
CURSOR RT X=
.8
SI 1
.242
F ig. 3 .5 .1 b
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
132
Normalized
T H eta=
Gate
Impedance
7 5 .0 0
p= . 2 0
CURSOR RT X=
0
SI 1
.2 4 3
F ig. 3 .5 .1 c
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
1 33
Normalized
T h e ta ;
Drain
Impedance
8
SI 1
5.00
p = 8 .0 0
CURSOR AT X
25 4
F ig . 3 .5 .2 a
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
13U
No r ma l i z e d
T h e ta =
Drain
Impedance
.8
ISI 1 -
4 5 .0 0
p= 8.00
CURSOR RT X=
F ig . 3.5.21:
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
135
Normalized
T h e ta =
Drain
Impedance
7 5 .0 0
co
p= 8 .0 0
CURSOR PIT X=
S
SI 1
F ig . 3 .5 .2 c
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
13 6
Gate Loss
Attenuation
o
.1
o
45
75
Fig. 3 .5 .3 .
Freq.
08/Div
.9
Gate lin e a tte n u a tio n v s. norm alized frequency fo r
th e case o f p=0.20 fo r v ario u s le n g th in d u cto rs ( th e t a ) .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
1 37
Drain Loss
Attenuation
in
>
•rH
Q
in
1
Freq.
08/Div
.9
45
F ig . 3 . 5 . 1*.
Drain lin e a tte n u a tio n v s. normalized frequency fo r
th e case o f p=8.0 fo r v ario u s length in d u cto rs ( th e t a ) .
R e p r o d u c e d with p e r m i s s io n of t h e c o pyright ow n er. F u r th e r re p r o d u c tio n prohibited w ith out p e r m is s io n .
138
th e g ate v o lta g e , th e fa c to r JZ0*S211 vas chosen to show th e combined
e ffe c t
o f changing input impedance and a tte n u a tio n .
The r e s u lt s
appear in F ig s . 3 .5 .5 and 3 .5 .6 .
What i s
lin e
most s tr ik in g is th e sm all changes which occur when
le n g th ,
th e ta ,
changes.
The obvious
a re tempered by th e decrease in
lin e
improvements in
in p u t impedance.
lo sse s
There is
a g en eral tr e n d toward b e tte r g ate lin e s as lin e length approaches
X/h (th e ta = 9 0 ).
In th e case o f th e d ra in l i n e , th e re i s a s l i g h t
decrease in q u a lity a t midband w ith in c re a s in g q u a lity fo r longer
in ducto rs above X=0.75, but th e d iffe re n c e between long and s h o rt
in ducto rs is n o t s ig n if ic a n t.
This d a ta
c u to ff
is
frequency
p o s s ib le .
somewhat r e s t r i c t e d
and
constant
input
as th e
case o f a co n stan t
impedance
is
not
g e n e ra lly
Changes in th e len g th o f a m ic ro s trip inductor a re u s u a lly
accompanied by changes in m ic ro s trip
impedance so as to m ain tain
th e lin e in p u t impedance o f th e sy n th esiz ed lin e a t some d e s ire d
v alu e.
This
r e s u lt s
in non-constant
c u to ff freq u en cies, and th e
a m p lifie r upper frequency lim it becomes a d if f e r e n t f r a c tio n , X_3^g
o f th e
c u to f f
frequency th an i t
was
d if f e r e n t le n g th m ic ro strip in d u c to r.
fo r each sy n th esis u sin g a
Because th e value o f X_3^g
is changing, th e r e s u ltin g lin e q u a lity f a c to r changes a t th e upper
frequency
order to
lim it
o f th e a m p lifie r.
The
im p licatio n
is
th is :
In
m ain tain input impedance c o n sta n t as th e len g th o f th e
m ic ro s trip in d u c to r in c re a se s, we r e q u ire lower impedance m ic ro s trip
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
139
Quality
.1/Div-
Gate Line
.1
a) 0 = 5°
b) 0 = 45°
c) 0 = 75°
%
F ig. 3.5.5*
Freq.
.08/Div.
Gate li n e q u a lity fa c to r v s. norm alized
fo r v ario u s in du cto r len g th s (t h e t a ).
frequency
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
1U0
Quality
.1/Div.
D rain Line
1
9
Freq.
08/Div
b) e = 45
c ) 8 = 75
F ig . 3 .5 .6 .
Drain lin e q u a lity f a c to r v s. norm alized
fo r various in d u cto r le n g th s ( th e ta ) .
frequency
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
in
order
to
fre q u e n c ie s.
and
fo rces
This
th e
a m p lifie r is
lin e
achieve
th e
loads
c u to ff
same
th e
e q u iv a le n t
lin e
frequency to
w ith
inductance
h ig h er
d e c re a se .
at
end cap acitan ces
Consequently,
forced to o p erate a t h ig h e r f r a c tio n s ,
c u to ff frequency which may r e s u l t
low
o f th e
in a d ecrease in
fa c to r w ith in c re a sin g m ic ro s trip in d u c to r le n g th .
th e
q u a lity
I t could a lso
r e s u l t in a maxima in th e q u a lity fa c to r f o r some p a r tic u la r in d u c to r
le n g th .
Whether or not t h i s
occurs depends on th e r a te a t which
X„3dB in c re a se s w ith in d u c to r le n g th , and th a t is t o t a l l y dependent
upon th e
tra n s is to rs
used
in th e a m p lif ie r , th e s p e c ifie d
in p u t
impedance, and upper frequency lim its .
F in a lly ,
we note t h a t
th e
phase d elay per s e c tio n
is a lso a ff e c te d by th e in d u cto r d esig n .
of
lin e
As th e frequency in c re a s e s ,
th e e f f e c t o f in c re a sin g end c a p a c ity i s
moderated by th e d evice
p a r a s itic ca p a c ita n c e s, C(js and Cgg, le a v in g th e inductor frequency
dependence to
dominate th e
phase v e lo c ity
o f th e
lin e .
As th e
e f f e c tiv e inductance d ecreases w ith in c re a s in g frequency, th e c u to ff
frequency
at
high
o f th e
lin e ,
freq u en cies
and hence
th e
in c re a sin g
in d u c to r le n g th , th e ta .
fo r
changes th e li n e a r it y o f th e phase d e la y .
li n e ,
phase v e lo c ity ,
in c re a se s
This
In th e case o f th e d ra in
shown in F ig. 3 .5 .7 , th e p o s itiv e cu rv atu re is reduced fo r
long in d u c to rs , and t h i s could be used to b rin g th e gate and d ra in
phase delays
in to b e t te r agreement.
3 .5 .8 )
an expected
shows
However, th e gate lin e
in c re a sin g n e g ativ e
curvature
fo r
(F ig .
long
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
a) e = 5°
o
b) © = 45'
o
c ) © = 75'
Fig.
3.5 .7.
9
Freq.
.08/Div.
D rain lin e phase delay v s . norm alized
v a rio u s in d u cto r lengths ( th e ta ) .
frequency
fo r
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
1U3
Gate Phase D elay
a
10 ° / D i v .
Phase
Delay
b
o
1
a) e = 5°
b) e = 45'
c ) 9 = 75'
F ig . 3 .5 .8 .
9
Freq.
•08/Div
Gate lin e phase d elay v s . norm alized
vario u s in d u cto r len g th s ( th e t a ) .
frequency
fo r
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
lkk
in d u cto rs which in d ic a te s th e use o f long in d u c to rs
in th e g ate
tran sm issio n
By ju d icio u s
lin e c i r c u i t may he o f lim ite d v alu e.
s e l e c t i o n 'o f th e m ic ro s trip in d u cto r len g th s
and impedances, and
hy s l i g h t l y sta g g e rin g th e c u to ff freq u en cies o f th e g ate and d ra in
li n e s ,
it
is
p o ssib le
to
optim ize
th e
a m p lifie r high frequency
respo n se.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
1^5
IV.
C o n strain t Free Device Design fo r S p e c ific A pplications
U .l.
Device param eter d e f in itio n from a m p lifie r s p e c if ic a tio n s .
In s e c tio n s
2 and 3, we developed a s e r ie s
o f curves which
could be used to g ra p h ic a lly so lv e fo r th e frequency response o f
an a m p lifie r u sin g a s p e c ific t r a n s i s t o r .
of
a
design
could
a m p lifie r re s id e d
be
stu d ied
by
of
examining
where th e
designed
in th e constant-K-X _3(jB cu rv es, which re p re se n t
norm alized gain-bandw idth product.
problem
A lso, th e r e la tiv e m erit
desig n in g
a
device
We w ill now study th e in v e rse
to
s u it
a
p a r tic u la r
design
s p e c if ic a tio n .
The most
common param eters
one
wishes
to
sp ecify a re
gain
(A0 ) , in p u t impedance ( R0 g) > ou tp u t impedance ( R0 ^ ) , and bandwidth
( f-3<jB) •
is
As we
a re designing th e t r a n s i s t o r we
n ecessary to a ls o
sp ecify th e d ev ice gain
d ev ices we w ill use (N).
wish to u se,
it
(gm) and how many
S ta r tin g w ith th e gain equation ( 1 .5 .6 ) ,
norm alizin g th e frequency, and s u b s titu tin g th e a and b c o e f f ic ie n ts
fo r otg and
as was done in Eq. 2 .1 .1 1 , we a r r iv e a t th e new gain
eq u atio n
A = j>m /R o g J W
2
s in h (a X2 / / l - gX2 - b / / 1 - X2 )
71 " X2 s in h [ ( a x2 / y i - gX2 - V / l - X2 )/N]
EXP[-(a X2 / >/l - gX2 + b /
A
v /l - X2 ) ]
(U .1.1)
+ X2 (2a/N )2
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Ih6
Evaluating this at X=0 gives the low frequency gain as
(it.1.2)
o r,
(U .1.3)
yRog-Rod
which i s
1-EXP(-2b)
a reasonable approxim ation fo r b< 0 .6 .
S im ila rly ,
from
Eqs. 2 .1 .8 c and 2 .1 .1 0 a one fin d s
Rg-Cgg =
2_
vN f c
( it.l.lt)
and from (2 .1 .8 d ) and (2 .1 .1 0 b ) we have
Rds'^ds =
Applying th e
(it. 1 .5)
Uirb f c
c o n s tr a in t
of
phase v e lo c ity
matching
on th e
gate
and d ra in lin e s , then
Rg = a Rog/N
( it.1.7)
(it.1.8)
and
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
lUT
tt.1.9)
Cds ;--------- —
^od
By u sin g th e s e e q u a tio n s, th e t r a n s i s t o r i s now s p e c if ie d .
The use o f th e s e
equations may be dem onstrated by designing
a h y p o th e tic a l a m p lifie r.
The a m p lifie r is re q u ire d to have 15dB
low frequency g ain , a 20GHz -3dB bandwidth, and 50 ohm input and
o u tp u t impedances.
th e
is
The a and b c o e f f ic ie n ts a re chosen to achieve
maximum gain-bandw idth which s e ts
little
p o in t to
a=0.75
and b=0.32.
There
doing o th erw ise, u n less a p a r tic u la r response
shape is re q u ire d .
At t h i s
a com pletely
determ ines
p o in t, th e choice o f device gm must be made.
I t is
fre e param eter, but i t must chosen w ith c a re ,
as i t
M which a ls o
a f f e c ts
Rg and Rd s .
Using (U .1 .3 ),
choose to s e t N=8 and so lv e fo r gm=38 mmhos.
F ig .
3 .2 .2 ,
at
s e ts f c=27GHz.
(a ,b )= (0 .7 5 ,0 .3 2 ), X » 0 .7 ^ .
we
From th e graph in
As f_ 3dB=20GHz, th i s
Based upon th e se v a lu e s, th e rem aining values are:
Cgs = 0.235pF
Rg = U.7 ohms
Cds = 0 .2 3 5 PF
Rds = 312 ohms.
The
tr a n s i s t o r
symmetry
in
is
now com pletely
CgS and
Cds
a r is e s
s p e c if ie d .
because
th e
The appearance
in p u t andoutput
impedances have been s e t eq u al to one an o th er.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
of
1U8
h .2
Physical constraints limiting device parameters.
The d iscu ssio n o f tr a n s i s t o r d esig n fo r s p e c ific a p p lic a tio n s
would he incomplete w ithout some c o n sid e ra tio n o f th e physics o f
th e
tra n s is to r.
It
is
not p o s s ib le
to a r b i t r a r i l y assig n v alu es
to th e model elem ents.
In th i s
d isc u ssio n , we w ill use th e more complete model o f
a MESFET shown in
F ig .
It.2 .1 .
In
Rs ,
Rg, Rfl, Rj, Cgs, and C^s w h ile
and
R(js .
nor
is maximum R(js : we simply d e s ir e
Minimum R j,
g e n e ra l, we wish to
sim ultaneously maximizing gra
Cgg, and C^s a r e not prima facea req u ire m en ts,
to
c o n tro l th e se param eters
so as to achieve th e goals o u tlin e d in S ect. U .l.
maximizing r e f le c t
minimize
g en eral d ir e c tio n s
M inimizing and
in which we ty p ic a lly w ill
d e s ir e to guide th e param eters.
F ig. U .2.1.
MESFET model.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
11+9
Schulz
[2l+]
summarized
th e
r e le v a n t
a n a ly tic a l
ex p ressio n s
for th e model element values:
c = ue q Nd
Es
: hulk co n d u c tiv ity
-
(1+.2.1)
: E le c tr ic f ie l d a t v e lo c ity s a tu r a tio n (X=Ll)
k = Es L/vp
:v e lo c ity s a tu r a tio n index
( U. 2 • 2)
g . g2
vp =----------2 e0e r
: p in c h -o ff v o ltag e
(h.2.3)
d = yv(L)/V p
= y(Vdg+VM )/Vp
(1+.2.1+)
:norm alized g a te -d ra in b ia s p o te n tia l
V-^ = 0.9V
: Schottky b a r r ie r p o te n tia l
s = \/-(Vgg-V-bi)/Vp
P
rnorm alized g ate-so u rc e
= /v d ^ J /V p
b ias
(1+.2.5)
(U .2.6)
» 1 fo r no v e lo c ity s a tu r a tio n (L]_ L)
*» s fo r extreme v e lo c ity s a tu r a tio n (Ld« L )
Cds = ( er +1)Eo W(K( 1/ 1-X2 )/K (x))
(U .2.T)
K(x) = e l l i p t i c in te g r a l o f th e f i r s t k ind.
f
X =
\ -i /p
(2Lg + Ld g ) L d g
( 1+. 2 . 8 )
S. ( L g + Ld g ) 2 ^
Cgd
= E0 £ r w (is + n/2)
(U .2.9)
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
150
Cgs
=
E0 £r w ( I & _ +
% = JLg
)
(It.2 .1 0 )
(1 - C (x ')/C g s )2 R (x ')d x '
( it.2.11)
tt/ 2
-as
R (x ') = [q ye (E(x'))N(i W b ( x ') ] ^
C (x ') =
wJ
£ 0 Er
0
(It.2.12)
dx"
a-b (x ")
(It.2.13)
gm = I § . l
(It.2.1M
I s = It E0er vs W/a
(U.2.15)
vs = s a tu r a tio n e le c tro n v e lo c ity
Rd = —— (LK(j+ciir/lr)
aaW °
(lr.2 .l6 )
Rds =
P V p /Is
(It.2.17)
Rs = — — (Lra + a(n/U - s ) )
a a W
&
(It.2.18)
R m g=777-
(It.2.19)
a Es
y
5 L,pZ
Since our design to o ls are based upon th e Rj/Cgg and Rdg-C^g
p ro d u cts,
we w ill
p e rip h e ra l i n t e r e s t
devote
in gm.
dependences, one finds th a t
most
of
our
From Eq.
a tte n tio n
It.2 . lit,
gm in c re a se s
to
them,
w ith
ev a lu a tin g geometry
fo r t h i n , wide d ev ices.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
151
H -X
o
Ll -*+** L l
Fig. 1 .2 .2 .
This
is
tr u e
w ell
as
devices
P lan a r MESFET channel p r o f ile .
for devices
w ith
-H
L
w ith alm ost no v e lo c ity
extrem e v e lo c ity
s a tu r a tio n
s a tu ra tio n .
as
T y p ic a lly ,
a
microwave t r a n s i s t o r o p erates c lo s e r to extreme v e lo c ity s a tu r a tio n
than to no v e lo c ity s a tu r a tio n .
The R(is‘cds product
re c ip r o c a l
c irc u it
is
device th ic k n e s s .
c h a r a c te r is tic
most s tro n g ly
dependent on 1 /a ,
As we wish to
frequency,
th e
in c re a s e d , and a th in dev ice does t h i s .
minimize th e
Rds'^ds
product
th e
d ra in
must
be
As R^s is b ia s dependent,
th e R ds'^ds product can be in c re ased by b ia sin g th e d evice c lo s e r
to p in c h -o ff.
The
Pi'Cgg
in t r a c ta b le .
product
is
th e
most
im portant
and
I t is very n e a rly independent o f w idth.
q u ite s ta b le w ith re s p e c t to dev ice th ic k n e s s , a .
th e
most
I t is
a lso
I t i s , however,
d i r e c t l y dependent on g ate le n g th : S hort gate len g th devices have
high g a te c i r c u i t c h a r a c te r i s t ic fre q u e n c ie s.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
152
It
is
not e n t ir e ly
c o rre c t to
lim it
th e
d isc u ssio n to
th e
R fC g s product sin ce th e model we a re u sin g lumps R-j_, Rs and R ^
in to an Rg'Cgg product.
not
fo r
Rs
to
p re se n t in th e
These a d d itio n a l term s in tro d u ce dependences
Rj/Cgg p ro d u ct.
v a r ia tio n s in g a te -to -s o u rc e
sp acin g ,
WhileR i’Cgg changes s lig h t ly
Lgg,
th e
presence o f
in th e Rg*Cgg product produces a term which responds d ir e c tly
Lgg.
Thus, we can modify, w ith in
li m it s ,
th e Rg'Cgg product
alm o st independently o f th e Rds *C^s pro d u ct.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
153
V.
Amplifier power considerations.
5.1
Power c a p a b ility o f d is tr ib u te d a m p lif ie r s .
So
fa r,
we have considered
sm all s ig n a l a m p lifie r.
a m p lifie r
is
ca n d id a te .
th e
d is tr ib u te d
a m p lifie r as
In many a p p lic a tio n s , a broadband power
req u ire d and th e d is tr ib u te d am p lifie r is
The
a
s p e c ific
power
c a p a b ility
of
th e se
a n a tu r a l
a m p lifie rs
re q u ire s c l a r i f i c a t i o n .
There
a re
s e v e ra l
lim itin g
fa c to rs
to
be co n sid ered .
cannot exceed th e d rain -so u rce breakdown v o ltag e o f th e FET.
gate
d riv e
v o lts ,
to
th e
th e g ate
is
FET is lim ite d to |Vp| +0.5 v o lts .
simply a forw ard
b iased Schottky
does not modulate d ra in cu rren t a p p re c ia b ly .
One
A lso,
Above 0.5
d io d e, and
Below Vp, th e p in c h -o ff
v o lta g e , th e d ra in c u rre n t is cu t o f f .
In
F irs t,
o rd er to
maximize power o u tp u t,
th e preceeding two co n d itio n s
two th in g s
must o ccu r.
must be met.Second,
th e
impedance o f th e d ra in lin e must be
Ro = (Vb r-Vs a t )2 /8P0
fo r maximum power o u tp u t.
it
i s o p erated class-A to
(5 .1 .1 )
As th e d is tr ib u te d am p lifie r is broadband,
p reserv e waveform f i d e l i t y :
Hence th e
fa c to r o f 8 in Eq. 5 .1 .1 a r is e s from th e quiescent d ra in b ia s o f
(V br~V sat)/2.
In a cascade ty p e a m p lifie r o p eratin g c la ss-A , we
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
15**
can so lv e fo r th e power, PQ, in term s o f th e g ate v o lta g e and d ra in
v o lta g e as
Po = gm(Vb r-Vs a t )(Vp+ 0 .5 )/8
(5 .1 .2 )
This eq u atio n expresses th e power handling c a p a b ility o f th e FET.
This
power
c a p a b ility
p erip h ery "
is
commonly expressed
in
term s
of
"g ate
or "g ate a r e a " , s in c e , in th e case o f s h o r t g ate len g th
microwave FET's,
th e
p erip h e ry
o f th e
g ate
is
n e a rly
a
lin e a r
fu n ctio n o f g a te w idth, and gm is a lin e a r fu n ctio n o f g ate w idth.
For
b ip o la r
tra n s is to rs ,
th e
corresponding term inology
is
"base
a rea" o r "base p erip h e ry ".
In th e
d is tr ib u te d a m p lif ie r ,
th e gain eq u atio n
(1 .5 .6 )
can
be w r itte n
VD =
(V 0>5) gmRod
e^V ^2
2 ^ 1 - f 2/ f c2 Vl + f 2/ f g2
Sinh(N(Yd “ Yg) / 2)
Sinh((Yd " Y g)/2 )
(5 .1 .3 )
Since
broadband
m u lti-o c ta v e
impedance
microwave
impedance d e s ire d .
d e s ire d v a lu e .
a m p lifie r
is
matching
a m p lif ie r s ,
The g ate
lin e
is
R0d
o fte n
is
not
u s u a lly
impedance i s
a ls o
f e a s ib le
set
set
fo r
to
th e
to
th e
This serv es two purposes: F i r s t , th e gain o f th e
lim ite d ,
c o n tro llin g g ain .
as
Rog and R0d a re th e
p r in c ip le
fa c to rs
Second, th e t o t a l power o u tp u t o f th e d is tr ib u te d
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
155
a m p lifie r is lim ite d .
This lim it e x is ts independent o f th e gatew idth
o f th e tr a n s i s t o r s used in th e a m p lifie r.
Thus, th e maximum power
outp u t o f a s in g le d is tr ib u te d a m p lifie r can not exceed t h a t given
by Eq. 5 .1 .1 .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
156
5.2
Comparison
of
the
power
gain
of
distributed
and
cascade
amplifiers.
In o rd er to e s ta b lis h th e r e l a t i v e m e rits o f d is tr ib u te d versus
cascade a m p lif ie r s , power gain was chosen as a b a s is .
power g a in ,
By s e le c tin g
e f f ic ie n c y can be s tu d ie d and th e tenuous n a tu re o f
arguments based on gain-bandwidth p ro d u ct and bandwidth alo n e can
be avoided.
1
F ig . 5 .2 .1 .
D is trib u te d a m p lifie r shown in gain-block form.
C onsider th e
5 .2 .1 .
N
block-form d is tr ib u te d
a m p lifie r
shown in
F ig.
When th e r e a re no lo s s e s ,
Vo = » gvd Vi = N
Vi Ro2 .
( 5 .2 .1 )
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
157
The power gain is
Gpd =
= N2g2d Ro1/Ro2 = R2gm2 r o1r q2
(5>2<2)
A s im ila r examination o f th e cascade am p lifie r y ie ld s
VQ =
Eve
(Jpc =
Ggj
Gpc
0
^i =
— —
Vi2/Ro l
_
Sm^o2 ^ i
= S^vcRo l / Ro2 = 8m^Ro lRo2
Bq1Bq2A = ^
(5*2.3)
(5.2.1*)
(5 2 5)
Em2 Ro lRo2
which s ta t e s th a t th e lo s s le s s d is tr ib u te d a m p lifie r re q u ire s two
devices to equal th e gain o f a s in g le t r a n s i s t o r cascade a m p lifie r.
When li n e lo sses a re accounted f o r , we can map
sinh(N (aK - ad )/2 ) e“N(a g+ad ) / 2
(
5 *2 . 6 )
sinh( (otg-ad )/2 )
This gives th e r a ti o o f th e power gains in term s o f th e e ffe c tiv e
number o f devices which is always le ss than N.
The power d is s ip a tio n
o f th e d is tr ib u te d a m p lifie r is
?d = N Id vds
(5 .2 .6 )
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
158
and for the cascade amplifier is simply
Pc = I d Vds .
(5 .2 .8 )
Thus, th e r a ti o is
Pd/P c = N
(5 .2 .9 )
T his, to g e th e r w ith
(5 .2 .5 )
and (5 .2 .6 ) s ta te s t h a t th e b est
d is tr ib u te d a m p lifie r w ill never be any b e tte r than h a lf as e f f i c i e n t
as a companion cascade a m p lif ie r .
This is not to be in te r p r e te d
as a blanket condemnation o f th e d is tr ib u te d a m p lifie r: I t o ffe rs
c le a r
advantages
c h a r a c te r i s t ic s .
in
What
bandw idth,
is
being
s ta b ility ,
s ta te d
is
th a t
and
th e
impedance
d is tr ib u te d
a m p lifie r is a r e l a t i v e l y low g a in , low e ffic ie n c y a m p lif ie r , and
th e u se r should be aware o f t h i s .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
159
5.3
C o n trib u tio n s o f in d iv id u a l d ev ices in th e d is tr ib u te d a m p lifie r
to o u tp u t power
In S e c t.
I,
th e equation
( 1 .5 .6 )
fo r gain o f a d is tr ib u te d
a m p lifie r was developed based upon th e prem ise th a t each t r a n s i s t o r
launches
two c u rre n t waves on th e
forward waves propagate along th e
th e o u tp u t te rm in a ls .
d ra in tran sm issio n
lin e
lin e .
The
and adds up in phase a t
This is an e n t i r e l y v a lid approach.
However,
simply because a device may develop a p a r tic u la r c u rre n t does not
imply a p a r ti c u la r power is
d e liv e re d by th e device to th e d ra in
lin e .
Because each tr a n s i s t o r e x c ite s two c u rre n t waves which t r a v e l
in o p p o site d ire c tio n s
on th e d ra in
stand in g wave e x is ts on th e d ra in li n e .
lin e a t a given frequency a
The am plitude o f th e v o ltag e
a t th e d ra in o f th e k -th t r a n s i s t o r is given by Eq. 5 .3 .1 .
k - ( l - l / 2 ) yfr -(k -£ )y d
6 e
+
vk “ vg zd( £ ee
1=1
.,
E e
I - k+1
(5 .3 .1 )
where
J l + X2 P_g2
J( 1-X2 )2 +U P_g2X2
(5 .3 .2 )
(5 .3 .3 )
1+ 2jP_d X
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
l 6o
£m — 1
X = f/fe
P S ~ 2Rg/RQg .
P_d = 2R,js /R qjJ
Vg i s
th e
th e
g ate
Z<j i s
c o n tro l v o lta g e o f th e t r a n s i s t o r
is
th e
at
a v -s e c tio n node on th e
ir-se c tio n
impedance o f th e
impedance seen a t a t r a n s i s t o r d ra in .
5 .3 .1
c u rre n t source when
g ate tran sm issio n
d ra in
li n e .
This
lin e .
is
th e
What has been done in Eq.
i s to sum th e forward waves on th e d ra in li n e w ith th e re v e rse
waves on th e d ra in lin e a t th e d ra in o f th e k - th d ev ice.
The phase
o f th e g ate v o lta g e due to th e reac tan ce s o f th e g ate lin e is common
to
b oth term s and has been removed from ( 5 .3 .2 ) .
gives o nly th e magnitude o f Vg(X) and t h i s
is
Thus,
(5 .3 .2 )
a l l we re q u ire as
a l l subsequent phase inform ation is d erived from i t .
I t is n ecessary to in clu d e both phase and magnitude inform ation
in
Z(j (Eq.
5 .3 .3 ) , as both a f f e c t power.
To c a lc u la te th e power
su p p lie d to th e d ra in l i n e , a l l th a t is n ece ssary is to s e t V^=l
and c a lc u la te V^-i^* where
*k = vg e- ( k - 1 /2 ^ g
(3 .3 .M
By exam ining th e d is tr ib u tio n o f power in th e a m p lifie r a t s e v e ra l
fre q u e n c ie s ,
one
can
gain
p h y sical in s ig h t
in to
v arious
asp ects
o f d is tr i b u te d a m p lif ie r s .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
l6i
Equation 5 .3 .1 can be used to show how Eq. 1.5*7 fo r th e optimum
number o f
devices a r i s e s . At some frequency th e
be driv en
ju s t hard enough
lin e
it
loss
in tro d u c e s .
l a s t d evice w ill
to compensate fo r th e
a d d itio n a l d rain
The adding o f an o th er t r a n s i s t o r
w ill
in tro d u ce lo s s .
This can be seen q u ite c le a r ly in F ig . 5 .3 .1 .
th is
a m p lif ie r ,
p a r tic u la r
a m p lifie r
perform ance.
th e
la s t
The drop
two t r a n s i s t o r s
in
d ra in
v o lta g e
In
degrade th e
at
th e
fifth
t r a n s i s t o r is due to th e in te rfe re n c e o f th e re v e rs e c u rre n t wave
from th e s ix th t r a n s i s t o r
fifth
tra n s is to r.
g e n e ra tio n .
one finds
C le a rly ,
By ap p ly in g
th a t,
and th e
at
th e H -field is n o t.
c u rre n t g en erato r.
forward c u rre n t wave from th e
th e re
is
no
u n ifo rm ity
boundary c o n d itio n s
a d ra in node, th e E - f ie ld
from
is
of
fie ld
power
th e o ry ,
co n tin u o u s, but
This is due to th e presence o f th e t r a n s i s t o r
What th i s
means is
th a t,
in
th e presence o f
th e o th er a c tiv e d ev ices on th e d ra in l i n e , th e impedance looking
in to th e d ra in connection to th e lin e w ill not be R0(j .
In s te a d ,
i t w i l l be th e complex r a t i o o f drain lin e v o lta g e to c u rre n t.
Another in t e r e s tin g asp ect to examine i s th e in h e re n t d isp e rsio n
between th e g ate and d ra in lin e s .
I f th e a m p lifie r is made long
enough, th e re
should be phase re v e rs a ls between th e
forward wave
on th e d ra in
lin e and th e d riv e on th e g ate l i n e .
This can be
seen
1*0 d ev ice am p lifie r
in th e
is
an
in
F ig .
ex ag g eratio n ,
5 .3 .2 .
it
Although a
n ic e ly
1*0
tra n s is to r
a m p lifie r
dem onstrates
th e p o in t.
Here, th e n eg ativ e value a tta c h e d to th e power developed
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
162
Watts/Div.
volts/Div.
D evice V o lta g e and Power v s . D evice #
6
50 Devices/Div
X = 0.8
Total Devices = 6
Overall Gain
a = -45
:
-.83 dB
b = .75
Fig. 5.3.1
Contributions of individual transistors in a
distributed amplifier vs. position in the amplifier.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
16 3
D evice V oltage and Power v s . D evice #
Volts/Div.
in
tn
Watts/Div.
1
3.90 Devices/Div.
X = '8°
Total Devices = 4 0
Overalll Gain: 8.25dB
a _ .49
b = .60
Fig. 5.3.2
Contributions of the individual transistors in
a distributed amplifier vs position in the amplifier for
a very long amplifier.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
l6k
by some tr a n s i s t o r s
th e d ra in
lin e .
in d ic a te s th a t th ey a r e e x tra c tin g power from
F ig .
5 .3 .3 shows th a t th e 9- th device is
q u ite w e ll driv en a t X=.8, but is
s till
in t e r f e r in g d e s tr u c tiv e ly w ith
th e s ig n a l a t i t s d ra in from th e o th e r t r a n s i s t o r s .
An in te r e s tin g asp e c t o f th e d is tr ib u te d a m p lifie r not tr e a te d
in th e
lite ra tu re
is changed.
is
th e
ev o lu tio n o f d ev ice power as frequency
Consider th e graphs in F ig s. 5 .3 .k and 5 .3 .5 .
These
show th e power d e liv e re d to th e d ra in lin e fo r th e f i r s t and l a s t
devices
of
a
tr a n s i s t o r s
th e
is
device
d is tr ib u te d
develop about th e
a m p lifie r
n e a rly
6
is
uniform
s h o rt
same power.
e le c tric a lly ,
from th e
a m p lif ie r .
firs t
to
At
This
is
and th e d ra in
th e
la s t
X = .l,
expected,
it
tra n s is to r
decreases
d isp la y s
r is in g
power o u tp u t
dev ice.
up to
s te a d ily w ith in c re a sin g frequency.
th e d riv e is a lso d e cre asin g , a l b e i t slo w ly .
as
lin e v o ltag e
The low
frequency r o l l - o f f is due to th e r i s i n g d ra in lin e re a c ta n c e .
s ix th
both
The
X=0.25 and
Note a lso th a t
Drive to th e f i r s t
tr a n s i s t o r is r is in g s te a d ily w ith frequency, y e t i t s power output
fa lls
ra p id ly a t
firs t,
rem ainder o f th e band.
and hovers between zero and 1/2 over th e
Obviously, th e assum ption th a t a d is tr ib u te d
a m p lifie r has i t s power uniform ly d is tr ib u te d among th e tr a n s i s t o r s
is not tr u e in g en eral a t any frequency.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
165
D evice Power and Current v s . Frequency
(N
Watts/Div.
.9
.1
.08 X/Div.
Device # = 9
a = .40
b = .60
Total Devices = 40
Fig. 5.3.3
Evolution of device power vs frequency for the
9-th transistor in a 40 transistor distributed amplifier.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Device Power and Current vs. Frequency
>
•H
Watts/Div.
Q
CO
a
in
o
.10
08 X/Div
.90
Total Devices = 6
Fig. 5.3.4
Power d eveloped by the last transistor of a
transistor distributed amplifier vs. frequency.
Note
that the generated power goes negative near X=.90
6
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
16 7
D evice Power and C urrent v s . Frequency
U1
Watts/Div.
Amps/Div.
m
-X.
o
.08 X/Div.
.90
Device # = 1
Total Devices = 6
Fig. 5.3.5
Power developed by the first transistor of a
transistor distributed amplifier vs. frequency.
Note
that while the drain current is rising, the generated
o ut p u t power is relatively constant.
6
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
168
VI.
Conclusion
In th i s t h e s i s , th e h is to r y and fundamental o p e ra tin g th eo ry
o f th e
d is tr ib u te d a m p lifie r have been p resen te d .
Equations
fo r
th e gain o f a d is tr ib u te d a m p lifie r in th e presence o f n o n -n e g lig ib le
lo sse s
These
in th e
lo sses
g ate and d ra in
were
c o e f f ic ie n ts ,
a
d escrib e d
and
b,
d is tr ib u te d a m p lifie r.
s e v e ra l
s e rie s
of
tran sm issio n lin e s were p resen te d .
in
norm alized
which were
then
term s
used
by
to
th e
design
an aly ze
th e
From th e normalized a m p lifie r g ain e q u a tio n s,
curves
d is tr ib u te d a m p lifie rs .
were
developed
as
d esig n
a id s
The f i r s t curves, th e co n stan t
fo r
curves
were used to p re d ic t
a m p lifie r gain and bandwidth fo r a d esig n er
s e le c te d t r a n s i s t o r .
The second s e r ie s o f curves were
gain-bandwidth cu rv es.
of
a th re e
curves
could
tr a d e - o f f s
tra n s is to rs )
to
These curves were shown to be p lan e s e c tio n s
dim ensional su rfa c e
c o e f f ic ie n ts , a and b ,
be
could
and
bandw idth,
be
and were d escrib ed
as were th e X_3,jb cu rv es.
overlayed
(g a in ,
th e co n stan t
examined.
a
lin e
complete
in
same
Thus, th e two
a n a ly s is
impedance,
th e
and
of
design
number
of
A design example was p resen ted
dem onstrate th e use o f th e se s e ts o f curves and how c i r c u i t
param eters ( i . e . ,
bandwidth,
and
Rog, R0<i, and f c ) and a m p lifie r resp o n se (g a in ,
gain-bandw idth)
-were
d ic ta te d
by
th e
chosen
tra n s is to r.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
169
It
was
a lso
noted
th a t
th e
c o n sta n t
gain-bandw idth
curves
had a maximum which corresponded to a p a r ti c u la r value o f th e design
c o e f f ic ie n ts ,
a
and
b.
This
maximum occured
gain-bandw idth o f one, and from t h i s
design
at
a
norm alized
i t was c le a r th a t an optimum
fo r th e d is tr ib u te d a m p lifie r e x is te d and i t
corresponded
to a p a r ti c u la r frequency response shape and a p a r tic u la r norm alized
bandwidth.
The re c ip r o c a l problem o f d esig n in g a custom t r a n s i s t o r
a s p e c if ic a p p lic a tio n was a lso p re se n te d .
for
The p h y sic a l lim ita tio n s
o f th e d esig n o f an a c tu a l t r a n s i s t o r were examined and suggestions
were made as to device param eters which could reaso n ab ly be v arie d
in o rd er to c o n tro l s p e c ific param eters ( i . e . , Rg , Cgs, R(js , C<js ).
The design o f tran sm issio n lin e in d u cto rs fo r MMIC d is tr ib u te d
a m p lifie rs was examined.
a
le n g th
lim it
of
For quasi frequency independent beh av io r,
X/T
was
su g g ested .
The
e f f e c ts
of
th e
n o n -n e g lig ib le end cap acity o f th i s ty p e o f in d u cto r on a m p lifie r
frequency
response
s ig n if ic a n t
is
longer
effect
than
was
on
X/7,
a lso
in v e s tig a te d
frequency resp o n se.
th e
in d u cto r
e x h ib its
and
found
When th e
to
have
lin e
a
length
frequency dependences
which d ecrease th e a tte n u a tio n o f th e c o n sta n t-k tran sm issio n lin e s
a t h igh freq u en cies and also a l t e r th e d is p e rs io n o f th e li n e .
The
examined.
lim itin g
case
of
th e
d is tr ib u te d
tr a n s i s t o r
was
also
I t could be seen th a t th e lumped element lin e s in th e
d is tr ib u te d a m p lifie r had a d e f in ite advantage over th e d is tr ib u te d
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
170
lin e s in th e d is tr ib u te d t r a n s i s t o r .
dependence o f th e
o f th e
v o ltag e a t a ir-s e c tio n node in th e
d is tr ib u te d
a t th e s e nodes r is e s
in p u t v o lta g e
fo r
a
This aro se from th e frequency
a m p lifie r.
as th e
d is tr ib u te d
a m p lifie r
li n e
Because th e impedanceo f th e li n e
c u to f f frequency is
to th e tr a n s i s t o r s
g ate
ten d s to
th an
approached, th e
f a l l o f f le s s r a p id ly
i t does
fo r
th e d is tr ib u te d
t r a n s i s t o r where no s im ila r lo ss com pensation e x is ts .
The s e n s i t i v i t y
o f th e d is tr ib u te d
components was a lso analyzed.
a m p lifie r to
The e f f e c t
o f Rg,
its
v ario u s
Cgg, R<is ,
and
C<js on gain c le a r ly showed th e s ig n ific a n c e o f Rg and Cgg to th e
high
frequency
were
seen
to
lo sses
be
frequency g ain .
th e
o f th e
a m p lif ie r .
c o n tro llin g
elem ents
S im ila rly , R,js and C,js
which
determ ined
low
A q u a lita tiv e a n a ly s is o f th e s e n s i t i v i t y o f th e
a m p lifie r to th e Rg'Cgg and R d s'cds products was made u sin g th e
c o n sta n t X_3,jb curves.
The v a l id it y range o f th e approxim ations to th e lo ss fu n ctio n s
was examined.
I t was discovered t h a t an upper lim it o f frequency,
r a th e r th a n an upper lim it on th e value o f th e
fu n ctio n i t s e l f
more a c c u ra te ly d escribed th e lim ita tio n s o f th e approxim ations.
The problem o f matching image param eter lin e s to r e a l sources
o f loads was tr e a te d .
c u to ff
frequency o f
I t was found th a t above 0 .6 to 0 .7 o f th e
th e
lin e s ,
th e presence
of
th e
c o rre c t
te rm in a tio n on th e gate lin e could in c re a s e th e a m p lifie r bandw idth.
I t was a ls o
found th a t th e m -derived 1 /2 -s e c tio n matching network
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
171
re q u ire d some m o d ific a tio n to provide a b e tte r match to lo s s y lin e s .
This was accom plished by changing th e m-value from 0 .6 to a sm aller
v a lu e .
L a s tly , th e power c a p a b ility o f a s in g le d is tr ib u te d a m p lifie r
was examined.
I t was shown th a t th e re is a maximum o u tp u t power
determ ined by d ra in lin e impedance and device d ra in -s o u rc e breakdown
v o lta g e .
The d is tr ib u te d a m p lifie r was compared to a cascade type
a m p lif ie r , and i t
never be as
power
was shown th a t th e d is tr ib u te d a m p lifie r could
e ffic ie n t
g en era tio n
in s id e
as a cascade a m p lifie r.
th e
d is tr ib u te d
The q u estio n
a m p lifie r
was
of
examined.
I t was found t h a t th e in d iv id u a l tr a n s i s t o r s do n ot uniform ly d riv e
th e d ra in
lin e
and th a t th e number o f tr a n s i s t o r s
which develop
th e most power d ecreases as frequency in c re a s e s .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
172
R eferences
1.
W heeler, H.A., "Wide-Band
IRE-27, 1939, pp. *+29-1*38.
A m plifiers
2.
P e rc iv a l, W.S., "Thermionic
1+60562, Jan. 1937.
3.
G inzton, E .L .,
H ew lett,
W.R., Jasb erg , J .H .,
HOE, J .D .,
" D istrib u te d A m p lificatio n ", P roc. IRE-36, 19I+8, pp. 956-969.
Valve
fo r T e le v is io n ,"
C ir c u its " ,
B r itis h
Proc.
p aten t
1+. P e t i t , J.M ., Pederson, D .O ., "D istrib u te d A m plifier Theory",
P roc. Symposium on Modern Hetvork S y n th esis, P olytechnic I n s t i t u t e
o f Brooklyn, 1952, p. 202.
5.
Payne, D.V., " D istrib u te d A m plifier Theory", Proc. IRE-1+1, 1953,
pp. 759-762.
6.
S c a rr, R.W.A., "D iscussion
P roc, IRE-1+2 , S951+, p. 596.
7.
B a s s e tt, H.G., K elley , L .C ., "D istrib u te d A m plifers:
Some Hew
Methods fo r C o n tro llin g Gain/Frequency and T ran sien t Responses
in A m plifiers having Moderate Bandwidths", Proc. IEE-101, p t.
I l l , 1951+, p. 5-11+.
8.
Sarma, D.G., "On D is trib u te d
1955, pp. 687-697.
9.
H orton, W.H., Ja sb e rg , J .H ., Hoe, J .D ., " D istrib u te d A m p lifiers:
P r a c tic a l C onsiderations and Experim ental R e su lts" , P ro c. IRE-38,
1950, pp. 71+8-753.
on
D istrib u te d A m plifier
A m p lificatio n ",
P roc.
Theory",
IEE-102B,
10. D eC laris, N., Shekel, J .
"Modern Hetwork S y n th esis Approach
to D istrib u te d A m p lifie rs" , 9 th IRE Southwestern E le c tro n ic s
Show and Conference, Houston, Texas, A pr., 1957.
11. M clver, G.W., "A
1965, pp. 171+7.
Traveling-W ave
T ra n s is to r" ,
12. Kopp, E.H ., "A Coupled Mode A nalysis o f
T ra n s is to r" , Proc. IEEE-5I+, 1966, p. 1571.
th e
P roc.
IEEE-53,
Traveling-W ave
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
173
13. J u t z i , W., "A MOSFET D is trib u te d A m plifier w ith 2 GHz Bandwidth",
Proc. IEEE-57, 1969, pp. 1195-1196.
lU . Schar, F ., " D istrib u te d MOSFET Amplifer u sin g M ic ro s trip " , I n t.
J . o f E le c tro n ic s , v o l. 3^, 1973, pp. 721-730.
15. S hapiro, L ., Y a., "A nalysis o f D iscrete-U niform S tru c tu re s used
in
D is trib u te d
A m p lifiers:
Telecommunications
and
Radio
E n g in eerin g , v o l. 29, #8, 1971+, pp. 95-101.
16. Chen, W.K., " D istrib u te d A m plification Theory", P roc. 5th Annual
A lle rto n Conf. on C ir c u it & System Theory, (U n iv e rsity o f
I l l i n o i s , Urbana), 1967, pp. 300-316.
17.
A y a sli, Y ., Mozzi, R .L ., Vorhaus, J .L ., Reynolds, L .D ., P ucel,
R .A ., "A M onolithic GaAs l-13GHz Traveling-Wave A m p lifier",
IEEE MTT-30, #7, J u ly , 1982, pp. 976-981.
18. A y a sli, Y ., Vorhaus, J .L ., Mozzi, R .L ., Hanes, L ., "Monolythic
2-20GHz Travelling-W ave A m p lifier", E le c tro n ic s L e tte r s , v o l.
18, n h , 8 J u ly , 1982, pp. 596-598.
19. Podgorski,
A .S .,
Wei,
L.Y .,
"Theory
of
Traveling-Wave
T r a n s is to rs " , IEE ED-129, #12, 1982, pp. 181+5-1853.
20.
S t r i d , E.W., G leason, R.K ., "A DC-12GHz M onolithic GaAs FET
D is trib u te d A m p lifier", IEEE MTT-30, #7, J u ly , 1982, pp. 969”975.
21.
Beyer,
J .B .,
P rasad ,
S .N ., Nordman, J .E .,
B ecker, R .C .,
Hohenwarter, G.K., "Wideband Monolythic Microwave A m plifier
S tudy", O ffice o f Naval Research re p o rt NR21+3-033-02, J u ly ,
1982 .
22. Op. c i t . ,
"Wideband M onolithic Microwave A m plifier
U n iv e rsity o f W isconsin, Dept, o f E le c tr i c a l and
E ngineering re p o rt ECE-83-6, S e p t., 1983.
Study",
Computer
23.
Ruston and Bordogna, E le c tr ic Networks,
and A n a ly sis, McGraw-Hill, 1966, S ect. 5.
F ilte rs ,
2h.
S chulz, J . , "R ela tio n o f th e Equivalent C irc u it to th e P h y sical
O peration o f MESFET's", MS T hesis, U n iv ersity o f W isconsin,
Madison, Aug. 16, 1983.
F u n ctio n s,
25. S ch ellen b erg , J.M ., Yamasaki, H., A sher, P .G ., "2 to 30 GHz
M onolythic D is trib u te d A m p lifier", 1981+ GaAs IC Symposium D ig est,
pp. 77-79.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
17U
Appendix I. Origin of the term 1/ <J1 - Xg^ in the gain express ion.
In t h i s appendix, i t w ill he shown t h a t th e term 1 / / l - Xg2,
where Xg = f / f Cg, c o r r e c tly d e sc rib e s th e behavior o f th e v o lta g e
p re se n t a t th e g ate o f th e FET's in a d is tr ib u te d a m p lifie r when
li n e lo sse s a r e n e g lig ib le .
o f term o r ig in a te s
I t w i l l a ls o be shown th a t t h i s type
in th e g ate l i n e fo r T -sectio n g ate and d ra in
l i n e s , and from th e d ra in lin e fo r ir-s e c tio n g ate and d ra in li n e s .
Lj
Lj/2
C js/2
F ig . A .I.
L ossless gate tra n sm issio n li n e showing it- and T -sectio n
n odes.
Consider
F ig . A .I.
v o lta g e a t
th e
co n stan t-k
tra n sm issio n
lin e
c irc u it
shown in
V^g is th e v o ltag e a t a T -se c tio n node, and V^g is th e
a ir-se c tio n node.
As th e
lin e
is
lo s s le s s ,
Vt g
and
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
175
L, / 2
♦
♦
Z ;s =*
Fig. A.2.
Vt
5
Cfs/2 ;
»R* V#5
A n a ly tic a l g ate tra n sm issio n lin e model.
virg re p e a t a t t h e i r re s p e c tiv e nodes.
I f th e segment o f th e lin e
shown in F ig . A.2 is analyzed, one fin d s th a t
V
virr
-
vts —
Vl-Xg2
L
---------
v/l-Xg2
+jXg
(A .l)
where
(A.2a)
Xg = io/<i)cg
“eg
= 2 / / y'g'cwgs
(A.2b)
Equation A .l can he re w ritte n in p o la r form as
V*g =
Vt g
(A .3)
-a rc ta n
L ^ x J
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
176
I t can a lso be shown th a t
~
Z^g = Rog
(A.ha.)
=^ tg
where
Rtg = ^og
J l “Xg^
(A.hb)
Rog = }/ ^g/^gs
which d efin e s
th e
(A.he)
T -sectio n
Eq. A.2a and A .3 th a t
in p u t
impedance.
It
is
c le a r
from
th e v o ltag e V^g on a T -sectio n lin e is a
r is in g fu n ctio n o f frequency.
I f one co n sid ers d riv in g th e gate li n e a t a ir-sectio n node,
one can e a s ily see th a t th e v o ltag e V^g w ill not be a r is in g function
of frequency.
This
is
im p lic it
in th e p e r io d ic ity
o f th e lin e .
As th e ir-se c tio n nodes are th e FET g ate te rm in a ls , th e T -sectio n
gate
lin e
c o n trib u te d
a
fa c to r J 1 - Xg^
to
th e
denominator o f
th e gain ex p ressio n .
Fig. A.3.
L ossless d ra in tran sm issio n lin e showing
n o d es.
it-
and T -sectio n
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
ITT
In
th e
case o f th e
d ra in
c irc u it
e q u iv alen t network is shown in F ig . A.h.
shown in
F ig .
A .3,
th e
This is a v a lid eq u iv alen t
only when th e lin e is uniform and pro p erly term in ated w ith an image
load.
Lrf/2
♦
4
v*4
" C * /2
•
F ig. A. h.
•R u
-
A n a ly tic a l d ra in tran sm issio n lin e model.
A fter some co n sid erab le a lg e b ra , one can show th a t
vtd = $ • Rod Z--arctan [ / = f
2
U l- X d2
(A .5)
and th e in p u t impedance, Z^d is
_
z id “
i
Rod
s
A -xd2
_ R
~ Rvd
(A.6)
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
1 78
where
^ird = ^od/ '/ l “ ^d^
(A.7a)
Rod = ^ V C d s
(A.Tb)
X<j =
m/ u C(j
(A.7c )
(A.Id)
“ cd = 2/
Note t h a t
V-td i s
frequency independent ( fo r lo s s le s s
li n e s ) .
If
one co n sid ers th e case o f a ir-s e c tio n d ra in l i n e ,
v*d = ^
R^d =
2
Thus,
1
2
( A .8 )
Vi-Xd^
a ir-sectio n d ra in
lin e
w ill
in tro d u ce a f a c to r
/ l - X^2
in to th e denominator o f th e gain ex p ressio n w hile a T -sectio n d ra in
li n e w i l l n o t.
In
o rd er to
c o r r e c tly
determ ine
a m p lif ie r , th e v o lta g e tr a n s f e r
must be accounted f o r .
( i.e .,
th e
gain
o f a d is tr ib u te d
fu n ctio n o f th e matching networks
For th e case o f symmetric topology a m p lifiers
both g ate and d ra in
lin e s
e ith e r T -se c tio n o r ir-s e c tio n ),
th e a n a ly sis is q u ite sim ple.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
1 79
AOO
f(X )
F ig . A .5.
S im p lified a n a l y tic a l a m p lifie r
matching elements (f( X ) ) .
C onsider
elem ents
th e
block-form
a m p lifie r
model
shown
in
w ith
F ig .
impedance
A.5.
The
f(x) re p re s e n t matching network tr a n s f e r fu n c tio n s .
If
we assume th e matching network to he r e c ip r o c a l, then
Vg = f(x) Vi
(A.?)
Vd = f(x) VQ
(A.10)
We th en can d escrib e th e a m p lifie r as
(A.11)
A =
Vg
The te rm
\ / l - X^ re p re s e n ts th e p ole introduced by th e g a te lin e
f o r T -s e c tio n to p o lo g ie s.
For ir-s e c tio n li n e s , th e term
r e p re s e n ts th e pole in troduced by th e d ra in lin e .
J l - X2
Using Eqs. A.?
and A .10, ( A .ll) becomes
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
180
Vd
yg
Vo
Vi
f(x )
f(x )
=
Thus, for symmetric topology lin e s , th e e f f e c t o f input and ou tp u t
matching networks d iv id es o u t, and th e raw a m p lifie r response given
by (A .ll) is th e response o f th e matched a m p lifie r.
The case o f asymmetric to p o lo g ies ( i . e , a ir-sectio n g a te li n e
and a T -se c tio n
d ra in
lin e ),
it
is
n ece ssary to
determ ine
th e
tr a n s f e r fu n c tio n o f th e matching network.
mL/2
“W W
l- m*i
2m
Vs
mC/2
I
F ig . A.6.
C ir c u it
fo r
determ ining
m atching
fu n ctio n s fo r T -sectio n l i n e s .
network
tr a n s f e r
The matching c i r c u i t shown in Fig. A.6 is fo r T -sectio n li n e s .
was shown in S e c t.
c a se , Vj = Vs / 2 ,
d iv id e r.
I,
when m=0.6, Z^ ssRq fo r X _< 0 .8 .
As
In t h i s
and th e problem reduces to a simple L-R v o lta g e
The t r a n s f e r fu n ctio n is then
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
181
Vt
Z_~tan' • i f °«
1^X 2
l-m^Tx^"
^
It
was p re v io u sly e s ta b lis h e d th a t
at
m=0.6.
A lso,
(A.13)
Eq.
as we a re in te r e s te d
A.9 would be v a lid
only
only in th e magnitude of
(A .13 ), we have
h
(A.1^)
Vi
as
th e
A .- ,,.* *
tra n s fe r
= ft(x)
fun ctio n
o f th e
matching network fo r T -sectio n
lin e s .
m L /2
Ro
— v/w —
v*(i=>
F ig . A.7.
*
• f
i
i
i
i
i
i
i
♦
If
11
♦
l- m l r
T
m
Vi
-
5 Tm C/2 ;
C irc u it
fo r
determ ining
matching
fu n ctio n s fo r tt- se c tio n s l i n e s .
R epeating
th e previous
procedure
fo r th e
V«
j
j n
1
network
tr a n s f e r
n -s e c tio n
matching
c i r c u i t , and ag ain using m=0.6 so t h a t Z j'R 0 fo r X _< 0 .8 , we find
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
182
>/i - o . 6 k x 2
Vi
as th e
/W C 2
tr a n s f e r
(A .15)
= M x)
fu n ctio n
o f th e
ir-se c tio n matching network.
We
can now examine th e asymmetric topology a m p lifie rs .
F irs t,
T -sectio n
consider
g a te
th e
lin e .
The
case
o f a n -s e c tio n
c irc u it
is
d ra in
rep resen ted
in
lin e
and
a
"block form
in F ig. A.8.
AtiO
Vs
rrr
F ig. A.8 .
C irc u it fo r determ ining th e tr a n s f e r fu n ctio n o f a
T -sectio n g ate l i n e , ir-se c tio n d rain lin e d is tr ib u te d
a m p lifie r.
From Eq. A.2 and A.8 , th e r e a re two poles in th e tr a n s f e r fu n c tio n .
We w rite
A=
A'
vird
A -X g2 J 1-Xd2
Vtg
(A.16)
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
183
where A’ i s
th e
gain o f th e
a m p lifie r w ith th e e f f e c ts
frequency dependence o f th e lin e s removed.
o f th e
Using ( A. l h) and (A.1 5 ),
we can re w rite (A .l6) as
Vo
Vi
Virdl f t W
_________ A_|_________________ A - x d2
vtgI f TT(x)
yi-Xg2
</l-xd2
/ l - 0.6J+xg2 / l - 0.6Uxd2
or
A'
V i'
C le a rly ,
(A .17)
A - o . e t a . 2 N /i-o .6i+Xfl2
(A.17)
is
always
le s s
th a n Eq. A.12 which is
th e
case
o f symmetric topology a m p lifie rs .
In th e case o f a rr-sectio n g a te lin e and a T -se c tio n d ra in
li n e , Eq. A .l6 becomes
A = A1 =
Vtd
Vtt,
(A.18)
and
Vo
Vi
Vtd
virg
fir(X)
= A' A - 0 .6 ta g 2 '/i- 0.61tx^2'
(A . 19)
'/i-X g 2
When Xg = Xd , th e gain fu n ctio n (A.15) is s lig h t ly le s s th an Eq.
A.12 fo r X _< 0 .8 .
Thus, th e r e is no s ig n if ic a n t b e n e f it to mixing
to p o lo g ie s .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
18U
Appendix I I .
Tables o f C o e ffic ie n ts o f C onstant F ra c tio n a l Bandwidth
In t h i s appendix, we p resen t ta b le s o f th e a and b c o e f f ic ie n ts
which r e s u l t
in a p a r tic u la r -3dB f r a c t io n a l bandwidth, X_3(jB as
d escrib ed
S e c t.
in
of N, th e
2 .1 .
These ta b le s
number o f devices
a re
p resented
fo r v alu es
in th e a m p lif ie r , which range
from
two to te n .
Also p re s e n t a re two o th e r
columns o f d a ta :
th e
D.C.
gain
and r a t i o o f -ldB frequency to th e -3dB frequency. . D.C. gain is
in flu en ced
only
by th e
b - c o e f f ic ie n t
as
th e
gate
lin e
lo s s e s
re p re se n te d by th e a - c o e f f ic ie n t d ecre ase to n e a rly zero as frequency
d e c re a se s .
The r a t i o o f f-idB
*-3dB ex p resses a measure o f th e f la tn e s s
o f th e resp o n se o f a d is tr ib u te d a m p lifie r which has been designed
around a p a r tic u la r s e t o f c o e f f ic ie n ts .
if
one is
to
cascade a m p lif ie r s ,
as
This is necessary knowledge
th e
choice o f c o e f f ic ie n ts
w ill determ ine th e u ltim a te bandwidth o f cascaded a m p lifie rs .
th e
c irc u it
frequency
That
d esig n er has so much c o n tro l over th e shape o f th e
response
is
one
of
th e
more
unusual asp ects
of
th e
d is tr ib u te d a m p lifie r.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
185
Constant Fractional Bandwidth Table
Nufiber of devices:
Fractional Bandwidth:
Error(dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1 .00
1.10
1 .20
B
1.088
.747
.544
.416
.328
.263
.209
.164
.125
.088
.055
.022
DC Gain
.39
SI
.60
.67
.73
.78
.82
.85
.88
.92
.95
.98
iber of devices:
ictional Bandwidth:
or(dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1 .00
1 .10
B
1.306
.775
.519
.372
.277
.209
.156
.116
.081
.050
.025
2
.90
.010
F(-ldB)/F<
.864
.853
.814
.742
.651
.569
.503
.452
.411
.377
.349
.326
3
.90
.010
DC Gain
.34
.50
.62
.70
.77
.82
.86
.89
.92
.95
.98
F <-idB>/F <
.845
.857
.842
.798
.727
.649
.580
.522
.476
.439
.408
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
186
Nunber of devices!
Fractional Bandwidth •
Error(dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1.00
B
1.475
.784
.506
.353
.252
.180
.125
.083
.047
.017
4
. 90
•010
Nunber of devices:
Fractional Bandwidth:
Error(dB)
A
.10
.20
.30
.40
SO
.60
.70
.80
.90
B
1.625
.788
.500
.341
.238
.163
.106
.063
.025
F<-idB)/F<
.831
.859
.853
.819
.762
.691
.622
.563
.515
.475
DC Gain
31
50
63
72
79
84
.88
92
.95
.98
5
90
010
Nunber of devices:
Fractional Bandwidth:
Error(dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
B
1.750
.794
.497
.334
.230
.153
.095
.050
.013
F<-id8)/F<
.819
.860
.858
.832
.781
.716
.648
.588
.S39
DC Gain
.29
.50
.63
.72
.80
.85
.90
.94
.98
DC Gain
.27
.50
.63
.73
.80
.86
.91
.95
.99
6
90
010
F(-idB)/F<
.810
.859
.861
838
.791
.730
.664
.604
.554
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
187
Nunber of devices:
Fractional Bandwidth
Error<dB>
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
B
1.850
.794
.494
.331
.225
.147
.088
.041
.003
7
90
•010
8
90
010
Nunber of devices:
Fractional Bandwidth
Error <dB>
A
.10
.20
.30
.40
.50
.60
.70
.80
B
1.950
.794
.494
.328
.222
.144
.083
.036
B
2.025
.794
.491
.328
.219
.141
.080
.031
F<-idB)/F(
.799
.861
.863
.844
.801
.744
.682
.622
DC Gain
.25
.50
.64
.73
.81
.87
.92
.96
9
90
010
Nunber of devices:
Fractional Bandwidth
Error<dB>
A
.10
.20
.30
.40
.50
.60
.70
.80
F<-idB)/F<
.804
.860
.863
.841
.797
.739
.675
.616
.565
DC Gain
.26
.50
.63
.73
.81
.87
.92
.96
1.00
DC Gain
.24
.50
.64
.73
.81
.87
.92
.97
F(-idB>/F(
.795
.861
.865
.844
.805
.749
.687
.628
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
188
Nunber of d e v i c e s :
F r a c t i o n a l Bandwidth:
Error(dB)
A
10
20
30
40
50
60
70
80
B
2.100
.794
.491
.327
.217
.139
.078
.028
10
.90
.010
DC Gain
.23
.50
.64
.73
.81
.87
.93
.97
F(-idB)/F(
.792
.862
.865
.84S
.807
.751
.690
.633
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
189
Constant Fractional Bandwidth Table
Nunber of devices:
Fractional Bandwidth:
Error(dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1 .00
B
1.788
1.275
.913
.663
.488
.356
.250
.163
.088
.022
DC Gain
24
34
44
54
63
71
78
85
92
98
Nunber of devices:
Fractional Bandwidth:
Error(dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1.00
B
2.425
1.475
.950
.656
.472
.338
.238
.153
.084
.025
Nunber of devices:
Fractional Bandwidth:
Error(dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1 .00
B
3.100
1.600
.963
.656
.463
.325
.222
.141
.072
.013
2
.80
.010
F(-idB)/F(-3dB)
.744
.734
.7iS
.686
.645
.598
.552
.509
.471
.438
3
.80
.010
Gain
18
31
44
55
64
73
80
86
92
98
F(-idB)/F(-3dB)
.731
.727
.726
.712
.683
.648
.608
.571
.534
.502
4
.80
.010
Gain
IS
29
44
55
65
73
81
87
93
99
F(-idB)/F<-3dB)
.723
.724
.730
.720
.699
.670
.636
.601
.567
.536
R e p r o d u c e d with p e r m i s s io n o f t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n proh ibited w ith o u t p e r m is s io n .
190
Nunber o f d e v i c e s :
Fractional Bandwidth:
Error(dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1.00
B
3.750
1.700
.969
.650
.456
.319
.216
.131
.063
.003
DC Gain
.12
.28
.44
.56
.65
.74
.81
.88
.94
1.00
Nunber of devices:
Fractional Bandwidth:
Error(dB)
A
10
20
30
40
50
60
70
80
90
B
4.450
1.750
.975
.650
.456
.316
.209
.125
.056
B
5.150
1.800
.975
.650
.450
.313
.206
.122
.053
F(-idB)/F<
.722
.720
.733
.727
.707
.681
.649
.618
.586
.556
6
.80
.010
DC Gain
.10
.27
.44
.56
.66
.74
.82
.88
.95
>r of devices:
ional Bandwidth:
(dB)
A
10
20
30
40
50
60
70
80
90
5
.80
.010
F(-ldB)/F(
.720
.720
.733
.728
.710
.687
.659
.628
.597
7
.80
.010
DC Gain
.09
.27
.44
.56
.66
.74
.82
.89
.95
F(-idB)/F(
.718
.717
.734
.730
.715
.691
.664
.634
.604
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Nunber of d e v i c e s :
F r a c t i o n a l Bandwidth:
Error(dB)
A
10
20
30
40
50
60
70
80
90
B
5.800
1.850
.975
.650
.450
.313
.206
.119
.050
B
7.200
1.900
.975
.650
.450
.309
.203
.116
.047
F<-idB>/F<-3dB)
.718
.715
.735
.730
.716
.693
.666
.639
.609
9
80
.010
Gain
.07
.26
.44
.56
.66
.75
.82
.89
.95
B
6.500
i.875
.975
.650
.450
.309
.203
.119
.047
Nunber of devices:
Fractional BandwidthError <dB>
A
10
20
30
40
50
60
70
80
90
.010
DC Gain
.08
.26
.44
.56
.66
.74
.82
.89
.95
Nunber of devices:
Fractional Bandwidth:
Error <dB>
A
10
20
30
40
50
60
70
80
90
8
80
F(-ldB)/F(-3dB)
.717
.714
.736
.731
.717
•695
.670
.641
.613
10
.80
.010
DC Gain
.06
.26
.44
.56
.66
.75
.82
.89
.95
F<-ldB)/F(
.716
.713
.736
.731
.717
.696
.670
.644
.614
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
192
Constant Fractional Bandwidth Table
Nunber of devices:
Fractional Bandwidth:
Error(dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
i .00
B
2.675
2. 063
1.513
1.088
.794
.575
.406
.263
.144
.038
iber of devices:
ictional Bandwidth
or(dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1 .00
1 .10
B
3.875
2.750
1.750
1.175
.825
.600
.425
.294
.188
.094
.013
2
.70
.010
DC Cain
.14
.20
.29
.39
.49
.59
.68
.78
.87
.96
F(-idB)/F<
.684
.673
.663
.652
.632
.608
.580
.552
.523
.496
3
.70
.010
DC Gain
.10
.16
.26
.38
.48
.58
.67
.76
.83
.91
.99
F <-idB>/F<
.681
.670
.666
.660
.652
.635
.618
.596
.572
.550
.528
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Nunber o f d e v i c e s :
F r a c t i o n a l Bandwidth:
Error(dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1.00
1.10
B
5.100
3.450
1.925
1.200
.838
.600
.431
.300
.194
.103
.025
DC Gain
.08
.13
.24
.37
.48
.58
.67
.75
.83
.90
.98
Nunber of devices;
Fractional Bandwidth;
Error(dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1 .00
1.10
B
6.300
4. ISO
2. 050
1.213
.838
.600
.431
.300
.194
.106
.028
4
.70
. 010
F(-ldB)/F<
.679
.667
.664
.665
.658
.648
.632
.614
.595
.575
.556
S
.70
.010
DC Gain
.06
.11
.23
.37
.48
.58
.67
.75
.83
.90
.97
F<-idB)/F <
.679
.665
.663
.667
.663
.654
.640
.625
.607
.588
.571
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Hufiber of d e v i c e s !
F r a c t i o n a l Bandwidth:
Error<dB>
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1.00
1 .10
B
7.500
4.850
2.150
1.225
.838
.600
.431
.300
.194
.106
.028
Nunber of devices:
Fractional Bandwidth;
Error(dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1.00
1.10
B
8.750
5.550
2.200
1.225
.838
.600
.431
.300
.194
.106
.028
6
.70
.010
Gain
05
09
22
37
48
58
67
75
83
90
97
F(-ldB)/F<-3dB>
.679
.663
.661
.668
.665
.657
.645
.630
.614
.596
.580
7
.70
.010
Gain
04
08
22
37
48
58
67
75
83
90
97
F<-ldB)/F <-3dB)
.678
.663
.662
.670
.667
.659
.647
.633
.618
.602
.586
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
195
Nunber o f devices-F r a c t i o n a l Bandwidth:
Error(dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1. 00
1.10
B
10.000
6.200
2.250
1.225
.838
.600
.431
.300
.194
.106
.028
DC Gain
.04
.07
.22
.37
.48
.58
.67
.75
.83
.90
.97
Nunber of devices:
Fractional Bandwidth:
Error(dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1.00
1.10
B
11.200
6.900
2.300
1.238
.838
.600
.431
.300
.194
.106
.028
8
.70
.010
F(-ldB)/F(-3dB)
.677
.663
.661
.671
.668
.660
.649
.636
.621
.604
.590
9
.70
.010
DC Gain
.03
.07
.21
.37
.48
.58
.67
.75
.83
.90
.97
F(-idB>/F(-3dB)
.677
.662
.660
.669
.669
.661
.650
.637
.623
.607
.592
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Nunber o f d e v i c e s :
F r a c t i o n a l Bandwidth:
Error(dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1.00
1.10
B
12.400
7.600
2.300
1.238
.838
.600
.431
.300
.194
.106
.028
10
.70
.010
DC Gain
.03
.06
.21
.37
.48
.58
.67
.75
.83
.90
.97
F(-ldB)/F<
.678
.662
.661
.670
.669
.662
.651
.638
.624
.608
.594
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
197
Constant Fractional Bandwidth Table
Nunber of devices:
Fractional Bandwidth:
Error <dB)
A
10
.20
.30
.40
.50
.60
.70
.80
.90
1.00
1.10
B
3.898
3.270
2.615
1.984
1.457
1.064
.773
.547
.361
.204
.066
DC Gain
07
10
15
21
30
39
so
60
71
82
94
Nunber of devices:
Fractional Bandwidth:
Error<dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1.00
1.10
1.20
B
5.754
4.719
3.605
2.484
1.660
1.172
.857
.631
.455
.311
.188
.081
2
.60
.001
F(-idB)/F <-3dB)
.648
.642
.634
.627
.618
.607
.593
.577
.560
.541
.522
3
.60
.001
DC Gain
.05
.07
.11
.18
.28
.38
.47
.S6
.65
.74
.83
.92
F<-ldB>/F(-3dB)
.647
.641
.634
.629
.626
.621
.613
.603
.591
.578
.565
.551
R e p r o d u c e d with p e r m i s s io n o f th e co p y rig h t o w n e r. F u r th e r re p r o d u c tio n proh ibited w ithout p e r m is s io n .
1 98
Nunber o f d e v i c e s :
F r a c t i o n a l Bandwidth:
Error <dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1 .00
1.10
1.20
1 .30
B
7.609
6.164
4.578
2.898
1.762
1.211
.885
.658
.483
.343
.225
.123
.033
Nunber of devices:
Fractional Bandwidth:
Error <dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1.00
1 .10
1 .20
1 .30
B
9.469
7.602
S. 539
3.258
1.820
1.230
.898
.670
.496
.357
.240
.141
.053
4
.60
.001
Gain
.04
.06
.09
.16
.27
.37
.47
.55
.64
.72
.81
.89
.97
F<-idB>/F<-3dB>
.647
.640
.634
.629
.629
.627
.621
.613
.604
.594
.583
.571
.560
5
.60
.001
Gain
.03
.05
.07
.14
.26
.37
.46
.55
.63
.71
.79
.87
.9S
F<-ldB>/F<-3dB)
.647
.640
.633
.629
.630
.629
.625
.618
.610
.601
.592
.582
.572
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
199
Nunber o f d e v i c e s :
F r a c t i o n a l Bandwidth:
Error(dB>
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1.00
1 .10
1.20
1 .30
B
11.320
9.039
6.484
3.578
1.859
1.242
.904
.676
.503
.364
.249
.149
.063
Nunber of devices:
Fractional Bandwidth:
Error <dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1 .00
1 .10
1 .20
1 .30
B
13.172
10.477
7.438
3.875
1.883
1.248
.908
.680
.507
.368
.253
.154
.068
6
.60
.001
Gain
.03
.04
.06
.13
.26
.37
.46
.55
.63
.71
.79
.86
.94
F<-idB)/F<-3dB>
.647
.640
.633
.629
.631
.630
.627
.621
.614
.606
.597
.588
.579
7
.60
.001
Gain
.02
.03
.06
.12
.26
.37
.46
.55
.63
.71
.78
.86
.93
F<-ldB)/F(-3dB>
.647
.640
.633
.628
.631
.631
.628
.623
.616
.609
.601
.592
.583
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Nunber o f d e v i c e s :
F r a c t i o n a l Bandwidth:
Error <dB)
A
.10
.20
.30
.40
SO
.60
.70
.80
.90
1 .00
1.10
1 .20
1 .30
B
15.031
11.906
8.375
4.156
1.898
1.252
.911
.682
.509
.371
.256
.158
.072
8
.60
.001
Gain
02
03
05
12
26
37
46
55
63
71
78
86
93
9
.60
Nunber of devices:
Fractional Bandwidth:
Error <dB>
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1.00
1 .10
i .20
1.30
B
16.891
13.344
9.313
4.406
1.910
1.256
.912
.683
.511
.373
.258
.160
.074
F(-idB>/F(-3dB)
.646
.640
.633
.628
.632
.632
.629
.624
.618
.610
.603
.594
.586
.001
Gain
02
03
05
11
25
36
46
54
63
70
78
86
93
F<-idB)/F<-3dB)
.646
.640
.632
.627
.632
.632
.630
.625
.619
.612
.604
.596
.588
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
201
Nunber o f d e v i c e s :
F r a c t i o n a l Bandwidth:
Error<dB>
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1.00
1.10
1.20
1.30
B
18.734
14.781
10.250
4.641
1.922
1.258
.914
.684
.512
.374
.260
.161
.076
10
.60
.001
DC Cain
.02
.02
.04
.10
.25
.36
.46
.54
.63
.70
.78
.85
.93
F(-idB)/F(-3dB>
.646
.639
.632
.627
.632
.633
.630
.625
.619
.613
.60S
.S97
.589
R e p r o d u c e d with p e r m i s s io n o f th e co p y rig h t o w n e r. F u r th e r re p r o d u c tio n proh ibited w ithout p e r m is s io n .
202
Constant Fractional Bandwidth Table
Nunber of devices:
Fractional Bandwidth:
Error<dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
i .00
i .10
1 .20
1 .30
B
5.727
5.109
4.461
3.781
3. 078
2.391
1.785
1.316
.963
.688
.461
.270
.102
DC Gain
.03
.04
.05
.08
.11
.17
.24
.33
.43
.53
.65
.77
.90
Nunber of devices:
Fractional Bandwidth:
Error(dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
i .00
i .10
i .20
1 .30
i .40
1.50
B
8.508
7.516
6.492
5.422
4.305
3.156
2.199
1.570
1.168
.883
.662
.484
.333
.201
.084
2
.50
.001
F<-idB)/F(-3dB>
.622
.619
.616
.612
.607
.601
.594
.587
.578
.568
.557
.545
.533
3
.50
.001
DC Gain
.02
.03
.04
.06
.08
.13
.21
.29
.38
.46
.55
.64
.73
.82
.92
F<-idB>/F<-3dB)
.622
.620
.617
.614
.610
.606
.603
.600
.595
.S89
.583
.S7S
.568
.559
.551
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
203
Nunber of d e v i c e s :
F r a c t i o n a l Bandwidth:
Error(dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1.00
1.10
1.20
1.30
1.40
1.50
1.60
B
11.281
9.914
8.500
7.031
5.453
3.805
2.445
1.684
1.246
.953
.732
.559
.412
.286
.176
.076
DC Gain
.01
.02
.03
.04
.07
.11
.19
.28
.36
.44
.52
.60
.68
.76
.84
.93
Nunber of devices:
Fractional Bandwidth:
Error <dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1 .00
1.10
1.20
1.30
1.40
1.50
1.60
1.70
B
14.063
12.313
10.500
8.609
6.578
4.391
2.602
1.742
1.285
.986
.766
.592
.447
.324
.217
.121
.034
4
.50
.001
F<-idB>/F(-3dB)
.622
.620
.617
.614
.611
.608
.606
.605
.602
.598
.593
.587
.581
.575
.568
.561
5
.50
.001
DC Gain
.01
.02
.02
.04
.06
.10
.18
.27
.36
.43
.51
.58
.66
.74
.81
.89
.97
F(-ldB)/F(-3dB)
.622
.619
.617
.614
.611
.608
.608
.607
.605
.601
.598
.593
.588
.583
.565
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
201*
Nunber o f d e v i c e s :
F r a c t i o n a l Bandwidth:
Error <dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1.00
1.10
1.20
1.30
1 .40
1.50
1.60
1.70
B
16.836
14.703
12.500
10.188
7.688
4.922
2.719
1.773
1.305
1.004
.783
.609
.467
.345
.238
.145
.060
Nunber of devices:
Fractional Bandwidth:
Error<dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1 .00
1 .10
1.20
1.30
1.40
1.50
1 .60
1.70
B
19.609
17.094
14.484
11.750
8.766
5.422
2.797
1.797
1.316
1.014
.793
.620
.478
.356
.252
.158
.074
6
.50
.QOi
Gain
.01
.01
.02
.03
.05
.09
.18
.27
.35
.43
.50
.58
.65
.72
.80
.87
.94
F(-idB)/F(-3dB)
.622
.619
.617
.614
.611
.608
.608
.608
.606
.604
.600
.596
.S92
.587
.582
.576
.571
7
.50
.001
Gain
.01
.01
.02
.03
.04
.08
.17
.27
.35
.43
.50
.57
.64
.71
.79
.86
.93
F(-idB)/F(-3dB>
.622
.619
.617
.614
.611
.608
.609
.609
.608
60S
.602
.598
.594
.590
.585
.580
.575
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
205
Nunber o f d e v i c e s :
F r a c t i o n a l Bandwidth:
Error(dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1.00
1.10
1.20
1.30
1.40
1.50
1.60
1 .70
1.80
B
22.391
19.484
16.484
13.328
9.844
5.891
2.859
1.813
1.326
1.021
.801
.627
.485
.364
.260
.167
.084
.008
DC Gain
.01
.01
.02
.02
.04
.08
.17
.27
.35
.42
.5u
.57
.64
.71
.78
.85
.92
.99
iber of devices:
ictional Bandwidth:
or<dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1.00
1.10
1.20
1.30
1.40
1.50
1.60
1.70
1.80
B
25.156
21.875
18.469
14.891
IQ.938
6.344
2.898
1.820
1.332
1.025
.805
.632
.490
.370
.266
.173
.090
.015
8
.50
.001
F<-idB)/F<
.622
.619
.617
.614
.611
.608
.609
.609
.608
.606
.603
.599
.595
.591
.587
.582
.578
.573
9
.50
.001
DC Gain
.01
.01
.01
.02
.04
.07
.17
.27
.35
.42
.50
,S7
.64
.71
.78
.85
.92
.99
F<-ldB)/F(
.622
.619
.617
.614
.611
.608
.609
.610
.609
.607
.604
.600
.597
.593
.588
.584
.579
.S7S
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
206
Nunber o f d e v i c e s :
F r a c t i o n a l Bandwidth:
Error(dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1. 00
1 .10
1.20
1.30
1.40
1.50
1.60
1.70
1.80
B
27.938
24.250
20.469
16.453
12.000
6.781
2.938
1 .828
1.336
1.029
.809
.635
.494
.373
.270
.178
.095
.020
10
.50
.001
DC Gain
.01
.01
.01
.02
.03
.07
.17
.26
.35
.42
.50
.57
.63
.70
.77
.84
.91
.98
F(-ldB)/F(
.62 c!:
.619
.617
.614
.611
.608
.609
.610
.609
.607
.604
.601
.597
.594
.589
.585
.581
.576
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
207
Constant Fractional Bandwidth Table
Nunber of devices:
Fractional Bandwidth:
Error<dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1.00
1 .10
1 .20
1 .30
1.40
1.50
1 .60
1 .70
B
8.903
8.280
7.628
6.944
6.241
5.519
4.769
4. 000
3.231
2.491
1.863
1.380
1.009
.716
.473
.262
.073
2
.40
.001
DC Gain
.01
.01
.01
.02
.02
.03
.05
.07
.10
.16
.23
.31
.41
.S2
.64
.78
.93
F(-idB>/F<
.604
.603
.602
.600
.S98
.595
.592
.588
.584
.579
.574
.569
.563
.556
.549
.541
.533
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
Nunber o f d e v i c e s :
F r a c t i o n a l Bandwidth:
Error<dB>
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1.00
1.10
1 .20
1 .30
1.40
1 .50
1 .60
1 .70
1.80
1.90
2.00
B
13.272
12.278
11.266
10.234
9.184
8.116
7.019
5.894
4.731
3.588
2.613
1.938
1.478
1.150
.897
.691
.517
.364
.227
.102
3
.40
.001
Gain
00
01
01
01
02
02
03
05
07
11
17
24
31
38
46
54
62
71
80
90
F(-ldB)/F<-3dB)
.604
.603
.602
.601
.600
.590
.596
.594
.591
.589
.586
.583
.580
.577
.573
.568
.564
.559
.554
.549
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
209
Nunber o f d e v i c e s :
F r a c t i o n a l Bandwidth:
Error(dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1 .00
1 .10
1 .20
1 .30
1 .40
1 .50
1 .60
1 .70
1 .80
1 .90
2.00
2.10
2.20
B
17.641
16.272
14.894
13.488
12.063
10.619
9.156
7.628
6 .044
4.450
3 .081
2.209
1.680
1.323
1.061
.852
.679
.531
.400
.283
.177
.080
4
.40
.001
DC Gain
.00
.00
.01
.01
.01
.02
.03
.04
.06
.09
.IS
.21
.28
.34
.41
.48
.54
.61
.69
.76
.84
.92
F(-ldB)/F <
.604
.603
.602
.601
.600
.599
.597
.595
.594
.592
.590
.589
.587
.585
.582
.579
.576
.572
.568
.565
.561
.557
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
210
Nunber o f d e v i c e s :
F r a c t i o n a l Bandwidth:
Error(dB)
A
.10
.20
30
.40
.50
.60
.70
.80
.90
1.00
1.10
1.20
1.30
1.40
1.50
1 .60
1.70
1.80
i .90
2.00
2.10
2.20
2.30
2.40
B
22.000
20.256
18.494
16 713
14.913
13.094
11.238
9.288
7.263
S. 163
3.400
2.359
1.783
1.413
1.141
.930
.756
.609
.482
.367
.265
.171
.086
.005
5
.40
.001
DC Gain
.00
.00
.00
.01
.01
.01
.02
.03
.05
.08
.14
.20
.27
.33
.39
.45
.51
.58
.64
.71
.78
.85
.92
.99
F<-ldB)/F<
.604
.603
.602
.601
.600
.599
.598
.596
.595
.593
.592
.592
.590
.588
.586
.584
.581
.578
.575
.572
.569
.566
.562
.559
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
211
Nunber of d e v i c e s :
F r a c t i o n a l Bandwidth:
Error <dB>
4
.10
.20
.30
.40
.50
.60
.70
.80
.90
1.00
1 .10
1.20
1 .30
1.40
1 .SO
1.60
1.70
1 .80
1 .90
2.00
2.10
2.20
2.30
2.40
B
26.369
24.250
22.094
19.938
17.763
15.550
13.281
10.938
8.425
5.819
3.625
2.453
1.844
1.459
1.185
.972
.798
.653
.525
.413
.313
.221
.138
.059
6
.40
.001
6a in
00
00
00
01
01
01
02
03
04
07
13
20
26
32
38
44
50
56
62
68
74
81
87
94
F(-idB)/F(-3dB)
.604
.603
.602
.601
.600
.S99
.598
.597
.595
.594
.594
.593
.592
.591
.589
.587
.584
.582
.579
.577
.574
.571
.568
.565
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e rm is s io n .
212
Nunber of d e v i c e s :
F r a c t i o n a l Bandwidth:
E rror(dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1.00
1.10
1.20
1.30
1.40
1.50
1.60
1 .70
1.80
1.90
2.00
2.10
2.20
2.30
2.40
2.50
B
30.738
28.225
25.713
23.163
20.594
17.988
15.325
12.550
9.588
6.400
3.813
2.519
1.881
1.488
1.211
.998
.824
.679
.552
.440
.341
.250
.168
.092
.020
7
.40
.001
DC Gain
.00
.00
.00
.01
.01
.01
.02
.02
.04
.07
.12
.19
.26
.32
.37
.43
.49
.55
.60
.66
.72
.79
.85
.91
.98
F<-ldB)/F(
.604
.603
.602
.601
.600
.S99
.598
.597
.596
.595
.594
.594
.593
.592
.590
.588
.586
.584
.582
.579
.577
.574
.572
.569
.566
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
213
Nunber o f d e v i c e s :
F r a c t i o n a l Bandwidth:
Error <dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1 .00
1.10
1.20
1.30
1.40
1.50
1.60
1.70
1.80
1.90
2.00
2.10
2.20
2.30
2.40
2.50
B
35.106
32.200
29.313
26.388
23.425
20.425
17.350
14.144
10.675
6.963
3.944
2.566
1.905
1.509
1.230
1. 014
.841
.69S
.569
.459
.360
.270
.188
.113
.043
8
.40
.001
DC Gain
.00
.00
.00
.00
.01
.01
.01
.02
.04
.06
.12
.19
.25
.31
.37
.43
.48
.54
.60
.65
.71
.77
.83
.90
.96
F(-ldB)/F(
.604
.603
.602
.601
.600
.599
.598
.597
.596
.595
.595
.594
.594
.593
.591
.590
.588
.586
.583
.581
.579
.576
.574
.571
.S69
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r re p r o d u c tio n prohib ited w ith o u t p e r m is s io n .
214
Nunber of d e v i c e s :
F r a c t i o n a l Bandwidth:
Error(dB)
A
.10
.20
.30
.40
.50
.60
.70
.80
.90
1.00
1.10
1.20
1.30
1.40
1 SO
1.60
1.70
1.80
1.90
2.00
2.10
2.20
2.30
2.40
2.50
B
39.475
36.194
32.913
29.594
26.256
22.863
19.375
15.738
11.800
7.488
4.056
2.594
1.923
1.520
1.241
1.026
.852
.707
.580
.470
.372
.283
.202
.127
.058
9
.40
.001
DC Gain
.00
.00
.00
.00
.01
.01
.01
.02
.03
.06
.12
.19
.25
.31
.37
.42
.48
.53
.59
.65
.71
.76
.82
.88
.94
F(-idB)/F(
.604
.603
.602
.601
.600
.599
.598
.597
.596
.595
.595
.595
.594
.594
.592
.590
.589
.587
.585
.582
.580
.578
.575
.573
.571
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
215
Nunber o f d e v i c e s :
F r a c t i o n a l Bandwidth:
Err or(dB)
A
.10
.20
.30
.40
SO
.60
.70
.80
.90
1.00
1 .10
1.20
1.30
1.40
1.50
1.60
1.70
1.80
1.90
2.00
2.10
2.20
2.30
2.40
2.50
2.60
B
43.825
40.188
36.513
32.800
29.088
25.300
21.400
17.313
12.888
7.975
4.131
2.613
1.938
1.532
1.248
1. 033
.859
.714
.590
.480
.381
.292
.211
.138
.068
.004
10
.40
.001
DC Gain
.00
.00
.00
.00
.01
.01
.01
.02
.03
.06
.12
.19
.25
.31
.37
.42
.48
.53
.59
.64
.70
.76
.82
.87
.93
1. 00
F<-idB)/F<
.604
.603
.602
.601
.600
.599
.598
.597
.596
.595
.595
.595
.595
.594
.593
.591
.589
.587
.585
.583
.581
.579
.577
.574
.572
.570
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
216
Appendix III.
Comparison
of
the
Distributed
Transistor
to
the
Podgorski
and
Distributed Amplifier.
The
Wei
[1 9 ],
a m p lifie r
d is tr ib u te d
is
to
th e
tra n s is to r,
lo g ic a l
in f in ite s im a lly
i n f i n i t e number o f dev ices.
as
d escrib e d
ex ten sio n
sm all
of
th e
d ev ices
by
MESFET d is tr ib u te d
(g a te
w idth)
and an
In this, s e c tio n i t w ill be shown th a t
th e eq u atio n fo r gain o f a d is tr ib u te d t r a n s i s t o r can be reduced
to
th e g ain eq u atio n fo r th e d is tr ib u te d a m p lifie r.
w ill a lso be shown th a t th e d is tr ib u te d t r a n s i s t o r
F u rth e r,
is
it
lim ite d to
m onotonically d ecre asin g gain w ith in c re a s in g frequency.
D
Jd Z'|
W
F ig . C .l.
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
21 7
We begin by examining th e t r a n s i s t o r shown in F ig . C .l.
is
a p p lie d to th e gate a t Z=0 and s ig n a l is tak en out o f th e d ra in
a t Z=W.
at
Drive
Z=0.
The g ate i s te rm in ated a t Z=W, and th e d ra in is term in ated
The g ate
tra n s m is sio n l i n e s ,
and d ra in s tr ip e s
re s p e c tiv e ly .
form th e
The
in p u t and ou tp u t
eq u atio n
fo r th e
v o ltag e
a t any p o in t on th e d ra in i s given by Podgorski [19] as
Vd(Z) = j Z I0 (Z')v£(Z,Z')dZ' +
I0 (Z')v£(Z,Z’)dZ'
(C.l)
(C.2)
Io (Z ’ )= \ 8mm vg e”712
(C.3)
(C.h)
(C.5)
As we a re p rim a rily in te r e s te d
in th e
forward gain from Z=0
to Z=W, we e v a lu a te Eq. C .l over t h i s range and g et
(e-r 2 ” _ e-YlW}
= £ gmm Zd Vg
(C.6)
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
218
N-1
F ig. C.2.
D is c re tiz e d d is tr ib u te d t r a n s i s t o r :
Z=W/N;
Snun,= Smm ^z.
Now c o n sid er th e segmented d ev ice shown in Fig. C.2.
As Eq.
C .l is being ev alu ated only a t Z=W, i t can be re w ritte n
Vd (W) = £ gmm' VgZd [
k=l
which is
th e
w ill d i f f e r
e" (Yl " Y2 )kAZ
upper approxim ation to
th e
(C.T)
in te g r a l.
As su ch ,
it
from Eq. 1 .5 .6 by an e x tr a 1/2 sectio n o f g a te lin e
and 1 /2 s e c tio n le s s o f d ra in lin e .
th e model shown is
F ig.
This occurs as Eq. C.7 assumes
C.3a re p re s e n ts
th e
c irc u it,
w h ile th e
model o f F ig . C.3b re p re se n ts th e a c tu a l c i r c u i t .
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
219
k- A2
-A
V z/2
92/ 2
Vz
Aa
A*
V i/2
V i/2
F ig. C .3.
Upper
Central
(a)
(b)
I n te g r a l approxim ation
d is tr ib u te d t r a n s i s t o r .
models
fo r
a n a ly s is
of
th e
When Eq. C.7 is summed and th e necessary lin e len g th c o rre c tio n s
a re made, th e r e s u lt is
Vd(W)= A
g^' z2e_N(Y1-1f2)AZ/2
X
sinh(NAZ(y1-Y2)/2 )
sinh(AZlY1-y 2 )/2
(C.8)
This d i f f e r s from Eq. 1 .5 .6 by a fa c to r o f 1 /2 , th e g a te lin e pole
(1 /
( f / f c g )2 ) ,
and th e
tran sfo rm a tio n
to
eq u al
impedances.
The e x tra fa c to r o f 1/2 comes about from Podgorski m easuring th e
g en era to r
open c i r c u it
v o lta g e ,
not
th e
v o ltag e
in to
a matched
load.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
220
The
absence
of
th e
fa c to r
1 / / l - ( f / f c g )2
is
th e
most
s ig n if ic a n t d iffe re n c e , as t h i s pole is re sp o n sib le fo r th e r is in g
gain
c h a r a c te r is tic
d e sc rib e d
lin e s
and low lo ss g a te l i n e s .
by Horton
[7 ]
fo r
lo s s le s s
drain
The lack o f t h i s term fo rces th e
g ain o f th e d is tr ib u te d t r a n s i s t o r to decrease m onotonically with
in c re a s in g frequency.
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
TITLE OF THESIS
C o n s tr a in ts in th e D esig n o f GaAs MESFET MMIC
D is t r ib u t e d A m p lifie r s _______________________________________________
MAJOR PROFESSOR
James B. Beyer________________________________
MAJOR DEPARTMENT Electrical Engineering________________________
MINOR
NAME
P hysics______________________________________ _______________
Robert C. Becker__________________________________________
PLACE AND DATE OF BIRTH
Madison, WI
15 June,
1954__________
COLLEGES AND UNIVERSITIES: YEARS ATTENDED AND DEGREES _______
University of
Wisconsin, Madison,
8/72 to 12/76
BSEE-]2/76
University of
Wisconsin, Madison,
8/79 to 8/80_____ MSEE-8/80
University of
Wisconsin, Madison,
8/80 to 5/85_____ Ph.D-5/85
MEMBERSHIPS IN LEARNED OR HONORARY SOCIETIES Member: Eta-Kappa-Nu,
Sigma Xi____________________________________________________________________
PUBLICATIONS "MESFET Distributed A m p l ifier Design Guidelines", J.B.
Beyer,
S.N. Prasad, R.C. Becker, J.E. Nordman,
G.K. Hohenwarter;
IEEE Transactions on Microwave T h eo r y and Technique, MTT-32, No.
1984, p p . 268-275_________________
________
3,
______________
DATE 4/29/85__________________________
F-5266
R e p r o d u c e d with p e r m is s io n of t h e cop y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
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