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Gallium-arsenide Monolithic Microwave frequency halver and phase detector for use in a phase locked oscillator

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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 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 .
Canada
- GaAs M o n o l i t h i c M i c r o w a v e
F r e q u e n c y H a l v e r and P h a s e D e t e c t o r
u q e i n a P h a s e -L o c k e d O s c i l l a t o r
for
by
SHAWN
A T h e s i s - s u b m i t t ed
R es ear ch fn p a r t i a l
STAPLETON
to the F aculty
f u l f i l l m e n t of
d e g r e e of
of G r a d u a t e S t u d i e s
t h e r e q u i r e m e n t s of
DOCTOR OF- P HI L OS OP HY
D e p a r t m e n t of E l e c t r o n i c s
F a c u l t y , of E n g i n e e r i n g
Carleton University
'
O t t a w a , Canada
June
19 8 7
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 .
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I SBN
0-315-46238-8
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 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 .
The u n d e r s i g n e d r e c o mme n d t o t h e f a c u l t y
and Reseasch a c c e p t a n c e of the -thesi s:
of Graduate
Studies
GaAs Monolithic Microwave
Frequency Halver and Phase Detector for
• H
use in a Phase Locked Oscillator v
s u b m i t t e d b y Sha wn S t a p l e t o n , BSc. E n g . , M . E n g . , i n p a r t i a l
f u l f i l l m e n t o f t he r e q u i r e m e n t s f o r t h e d e g r e e o f Doct oC^pf
Philosophy.
’*
P r o f > J \ S. Wi ght "
Thesis Supervisor
A
A
i
.
\
%
Prof.
M. A. ‘ Copel and
Ch a i r ma n ,
Department of E l e c t r o n i c s
Dr . M. S t u b b s
Commu n i c a t i o n s . R e s e a r c h C e n t r e
E x t e r n a l Ex a mi n e r
Carleton University
A p r i l 198 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 .
• ABSTRACT
*
i
-•
\
Ga As
( MMI Cs )
Monolithic
are
Mi cr owave , I n t e g r a t e d
Circuits
a
replacing
existing, technology
due
to
rapidly
*:v\.
their
Many
advantages
of
l;he.
attenuators,
basic
etc.)
oscillator's
arily
in---' s i z e ,
are
ta r e
of
cost
MMlC • c o m p o n e n t s
presently
being
because'
weight,
rejected
their
, r e 1 i a b i I i*t y
(mixers,
being
in
and
used.
ma n y
inherently
#
amplifiers,
Ho w' e v e r ,
applications,
poor
ist ics.
noise
MMIC
prim­
character**‘
___
•
♦
of
A. t e c h n i q u e
for
oscillators
by
MMIC
this
thesis.
Four
oscillator
to
•s o l u t i o n s .
a
comprised
of
d’i v i d e r s ,
loop'
c on p o n e n t s ,
hf f v§
no
of
tljese
and
only
the
case
Ritz
a
of
state
Their
introduced,
locking
a.noisy
as
viable
configurations
phase
frequency
detectors.
Of
and
phase
these
d e t e*c t o r
MMI Cs ,
halver
circuit
and
configurations
pha^se
detector.
analyzed
in
great
in
Bot h
detail
revealed.
algebraic
differential
of
method,
based
equation,
\
the
de t e c t o r , ' t h e
novel
detector
performance
theoretical
frequency
analysis
is
on
the
used to
*
halver.
is
solu­
obtain
In
the
b a s e d 'on
the
circuits
are
is
'for
designed,
then
evaluated
the
frequency
fabricated'
and
halver
and
compared
with
predictions.
*
and
measured.
s I
3ue.kL.,t.‘. - . .
ar e-
multipliers,
divider
novel
criteria
l o c k i ’ng
Method.
The
the
are
theoretically
operation
phase
phase
phase
in
frequency
approximate
the
and
presents
a nonlinear
for
oscillators,
realisation
operating
steady
phase
frequency
circ uits^ are
An
of
the
thesis
for
their
tion
different
filters
current
form
reference
mixers,
This
MMIC
configurations
clean
T. hes g
i mp r o v i n g the n o i s e c h a r a c t e r i s t i c s
•»
*
us e of p h a s e l o c k i n g , i s r e p o r t e d
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 .
their
ACKNOWLEDGMENTS
* *
t
*
*
I
The'
author
*
supervisor,
support
wishes
Professor
during
the
to
course
thesis.
■
' I
Wight,
^of
for
the
gratitude
his
wor k
to
his
guidance
reported
and
in
t ’h i s
"
The.
Dr.
■
J.S.
0
express’ his
M. G.
author
Stubbs
Vo c o u n t e r
in
privileges
for
a
Many
The
to
Research
thanks
extremely
launching
pub.
extended,
*
\
Communications
is
are
this
thesis
author
him
by
to
also
Dr .
after
Mr s .
M. G.
W.
a
to
brief
recognizes
Stubbs
J
Centre.
due
appreciative.
Brydges
t ‘h e
and
the
*
*
for
expertly
a
typing
this
Th e
the
■g a v e
staff
tlveir
the6i"s.
^
y
a u t h o r woul d
at
the
time
his
'
to 1e x t e n d
Communications
gratitude
the
to
author
Research
his
ik lo s t
parents,
for
over
.
appreciation
his
d u r i n g ' t h e i r v]d i s c u s s i o n s
Finally,
ing
like,
Ce n t r e 'who
with
words
all
to
fully
him.
wh e n
these
i
express­
past
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 .
y e a r s ..
TABLE
OF
. J/
CONTENTS
*
ABSTRACT........................................................
..................... '.........................................................
ACKNOWLEDGMENTS
TABLE O F ‘. CONTENTS........................................................................................
LI S T OF F I G U R E S ........................................
L I S T OF SYMBOLS.....................................................................
> L I S T OF TABLE.................................................................................................
CHAPTER
^
CHAPTER
1
I NTRODUCTI ON. ..............................................................................
1
1. 1
G e n e r a l .......................
1 . 2 T h e s i s O b j e c t i v e s ................................................ . ..............
1 . 3 T h e s i s O r g a n i z a t i o n ............................................................
*
2
GaAs MONOLI THI C MICROWAVE I NTEGRATED
C I R C U I T S ...........................................................................................
1
5
5
2.1
I
CHAPTER
8
2.2
2.3
2.4
P r o c e s s i n g - T e c h n i q u e s .....................................................
2 . 1 . 1 I n t e g r a t e d C i r c u i t T e c h n o l o g y ..............
2.1.2 Field Effect Transistor Fabrication
T e c h n i q u e s , , . ...........................................................
2.1.3 Capacitor Technology
. . .
2 . 1 . 4 I n d u c t o r T e c h n o l o g y ...........................................
2 . 1 . 5 R e s i s t o r T e c h n o l o g y .........................
D e s i g n M e t h o d o l o g y ...................................................... ..
Y i e l d C o n s i d e r a t i o n ......................................
GaAs MMIC A u t o m a t i c T e s t E q u i p m e n t S y s t e m
12
16
20
23
26
29
30
3
PHASE LOCKED OS CI L LATOR.................... ' . ........................
34
3.1
3.2
S y s t e m C o n f i g u r a t i o n ........................................................
S y s t e m C o m p o n e n t s ...............
3 . 2 . 1 O s c i l l a t o r ....................................................................
3 . 2 . 2 Low p a s s ' F i l t e r . ...................................................
3 . 2 . 3 M i x e r ............................................................................
3 . 2 . 4 M u l t i p l i e r ....................................................................
3 . 2 . 5 F r e q u e n c y D i v i d e r ................................................
3 . 2 . 6 B a i u n ...................................................................................
3 . 2 . 7 P h a s e D e t e c t o r ................................................... .
34
37
37
43'
44
45
47
50
52
3.3
CHA'PTER 4
4.1
4.2
4.3
4.4
Discussion
Component s
of
MMIC P h a s e L o c k e d L o o p
.......................................................................
k
.
56
58
T t e q u e n c y H a l v e r M o d e l .....................
F o r m u l a t i o n o f D i f f e r e n t i a l E q u a t i o n ...........
A n a l y t i c a l S o l u t i o n . , , .....................
A n a l y t i c a l S o l u t i o n to the Steady S t a t e
Response
.................................................
60
62
71
* „
%
—
■ 8
8
FREQUENCY H A L V E R . . . ....................................... , ...................
-
►I 0
I—
I
II
H i
IV
V
VI
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 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 .
83
Page
«
9
\
•
'
.
QHAPTER 5
5.1
5.2
5.3
5.4
CHAPTER 6 '
PHASE' DETECTOR
91
r
t
P h a s e D e t e c t o r M o d e l .......................
91
F o r m u l a t i o n of D i f f e r e n t i a l E q u a t i o n . . . . . .
94
A n a l y t l e a l ^ S o l u t i o n ..................................
95
O p t i m i z a t i o n of Phase D e t e c t o r
112
S e n s i t i v i t y ..............................................
LABORATORY M E A S U R E M E N T S . . / . .........................................
116
t
6.1
6.2
6.3
CHAPTER 7
7.1
7.2
F r e q u e n c y H a l v e r .........................................................................
6 . 1 . 1 C i r c u i t D e s i g n ....................
6 . 1 . 2 F r e q u e n c y H a l v e r M e a s u r e m e n t s ..................
6 . 1 . 3 C o m p a r i s o n of T h e o r e t i c a l and
E x p e r i m e n t a l R e s u l t s ............................................
P h a s e D e t e c t o r ...............................................................
6 . 2 . 1 C i r c u i t D e s i g n .............................................................
6 . 2 . 2 Phase D e t e c t o r M e a s u r e m e t i t s . . . . . . . . .
6 . 2 . 3 C o m p a r i s o n o f T h e o r e t i c a l and
E x p e r i m e n t a l R e s u l t s ............................................
C o n c l u s i o n s .....................................................................
117
1 17 v
121
135
139
. ACHI EVEMENT AND RECOMMENDATI ONS FOR
FURTHER RESEARCH.........................................................................
142
124
130
130
134
V
A c h i e v e m e n t ................................................................................. *. . 1 4 2
R e c o m m e n d a t i o n s f o r F u r t h e r R e s e a r c h . . . . . . ’ 143
REFERENCES...^*
. . *
.
\
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 .
144
-
LIST
OF
FIGURES
v
Page
v
2.1
P l a n a r GaAs C i r c u i t ' F a b r i c a t i o n P r o c e s s
Using' Ion I m p l a n t a t i o n D i r e c t l y i n t o Q u a l i f i e d 3
S e t a i - 1 n s u 1a t i n g S u b s t r a t e M a t e r i a l A d o p t e d
b ^ R o c k we 1 1 ..................................................................................................
11 '
Processing Steps
Technology
of
Self-Aligne-d Gate
. . ............................................................ ..
15
2.3
Processing
of
Etched- Channe1 Technology..
15
2 .4
T h r e e I m p l e m e n t a t i o n s o f C a p a c i t o r s on MMI Cs :
(a) a st ub in m i c r o s t r i p t r a n s m i s s i o n l i n e ;
( b ) an i n t e r d i g i t e d c a p a c i t o r ; ( c ) a m e t a l - i n s u 1 a t O T - m e t a I ( MI M) c a p a c i t o r ...........................................
18
2.2
2.5
2.6
Steps
E q u iva'L£j3tf* c i r c u i t o f an i n t e r d i g i t a t ed
capacitor.
Th e s h u n t c a p a c i t a n c e i s t o t h e
b ‘a &
c k s i d e g r o u n d p l a n e ..................................... .................................
.
^
18
^
E q u i v a l e n t ' c i r c u i t of a m e t a l - i n s u l a t o r - m e t a l
( MI M) c a p ac i t o ' r ............................ ..........................................................
20
2.7
Lu mp e d
-............
21
2.8
Two b a s i c t y p e s o f t r a n s m i s s i o n l i n e s u s e d on
MMI Cs :
( a ) c o p l a n a r ; ( b ) n i c r o a t r i p . . ....................
23
MMIC r e s i s t o r c o m p o s e d o f ^ c t i v e GaAs m a t e r i a l ;
( a ) d e f i n e d by . q e s a e t c h i n g ; ( b ) d e f i n e d by
i m p l a n t i s o l a t i o n or s e l e c t i v e i m p l a n t a t i o n . . . .
24
2.9
inductor
c o n f i g u r a t i o n s ................
\
2.10
2.11
R . F . M e a s u r e m e n t S y s t e m H a r d w a r e ..............
5S?
M e a s u r e d S - P a r a m e t e r s o f F i r s t 50 I . C . s o n
Wafer
................................................. ..........................
On-wafer
t ‘
*
32
33
*
C o n f i g u r a t i o n s .......................
36
R e s o n a t o r . . * . . . ; . .......................................................
42
Two T y p e s o f I n j e c t i o n L o c k i n g :
(a) R e f l e c t i o n
t y p e ; ( b ) T r a n s m i s s i o n t y p e , ...................................................
46
G e n e r a l Concept of R e g e n e r a t i v e F r e q u e n c y
D i v i s i o n ............................................................................................................
46
3*5
'Frequency
48
3.6
Transmission
3.1
Phase
3.2
Dielectric
3^3
3.4
Locked
Oscillator
D i v i d e r ....................... .. .........................................................
Line
B a i u n . . . . . . .................................................
IV
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 .
4 9ei
n
\
<s
Page
; ...........................
50
C o n f i g u r a t i o n s ............... ........................................... ..
5I
3,7
Monolithic
Microwave
3.8'
FET M i x e r
3.9
Diode-ring
3.10
Diode-Rect i f i e r
Phase
B aiun..,,
53
D e t e c t o r ........................................................ ...
Phase
D e t e c t o r .......................................
53
Diode
M i x e r .....................................................
55
P>
3.11
Single-Balanced
3.12
Double-FET
D e t e c t o r ...........................................................
55
4.1
Field Effect T r a n s i s t o r Frequency Halver;
( a) P h y s i c a l c i r c u i t l a y o u t ; (b) E q u i v a l e n t
c i r c u i t l a y o u t ...........................................................................................
61
Field
Nodal
Effect T r a n s is t o r Frequency Halver
V o l t a g e a n d C u r r e n t I d e n t i f i c a t i o n ............ ...
64
Boundary Region of u n c o n d i t i o n a l f r e q u e n c y
h a l v i n g as a f u n c t i o n of t h e s e r i e s r e s i s t a n c e
87
4. 2^
4.3
4.4
4.5
4.6
4.7
4.8
5.1
5 ."2
5.3
Phase
Boundary Region of u n c o n d i t i o n a l f r e q u e n c y
h a l v i n g a,8 a f u n c t i o n o f t h e g a t e t o s o u r c e
v o l t a g e . , , , ............ .. ..................................... ..............................................
87 ‘
Boundary Region of u n c o n d i t i o n a l . f r e q u e n c y
h a l v i n g as a f u n c t i o n of t h e f e e d b a c k
c a p a c i t a n c e ............... .................................................... .. ...........................
88
V o l t a g e g a i n v e r s u s h a l v i n g f r e q u e n c y as a
f u n c t i o n o f t h e l o q d r e s i s t a n c e . . . . . . . . ..................
’s
V o l t a g e g a i n v e r s u s h a l v i n g f r e q u e n c y as a
f u n c t i o n o f t h e g a t e t o s o u r c e v o l t a g e .....................
V o l t a g e g a i n v e r s u s h a l v i n g f r e q u e n c y as a
f u n c t i o n o f t h e f e e d b a c k c a p a c i t a n c e . . . . . ............
( a ) P h a s e D e t e c t o r ; (b)
Equivalent Circuit..,.
Phase
Defector's
..............................
P h a s e D e t e c t o r O u t vp u t - V o l t a g e s f o r V a r i o u s
B i a s i n g C o n d i t i o n s . . ........................... .. .................
P Phase
Detector
Sensitivity
Divider
Circuit
to
Bias
Conditions.
D i a g r a m . . . ...................
89
9
89
1
90
93
114
115
6.1
Frequency
6.2
Fabricated
Frequency
H a l v e r ....................................
119
6.3
Monolithic
Frequency
Divider
1 22
Circuit
Diagram..
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 19
♦'
Page
6.4
Fabricated
6.5.
Frequency
Divider
6.6
Region
of
Frequency
Division
6.7
Frequency
Divider's
Conversion
6.8
S we p t
Frequency,
Response
for
PxN “
6.9
T h r e s l f o l d Power
F r e q u e n c y ..............
Required
for
Division
6.10
Monolithic
v
»
Frequency
Measurement
D i v i d e r .........................
122.
A p p a r a t u s . , .......................
. . . .
123
125
G a i n ................................. ' . . 125
2.0
dBm.......
against
.....
126
. 129
H a l v e r O u t p u t S p e c t r u m f o r S we p t 17 dBm
I n p u t **SJL g n a ' i . ' \ .........................................................................
v
129
*
*«
6.11
Phase
D ejfe^ftr.
Using
6.12
Equivalent
Phase
6.13
Fabricated
Monolithic
6.14
Phase
6.15
F E T s ............................................................
Detector
C i r c u i t ..............................................
133
D e t e c t o r ......................................
133
A p p a r a t u s . . . * .............................
134
P h a s e D e t e c t o r C h a r a c t e r i s t i c s as a F u n c t i o n
o f V o l t a g e a n d P o w e r . . .............................................................................
138
Detector
Phase
132
Measurement
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 .
\
LIST
SYMBOL
OF
SYMBOLS
FREQUENCY HALVER
’
«
Q
Quality
factor
V
Re
Series
Rl
Load
Vp
Instantaneous
(D0
Small-signal
A
June t i o n a r e a
e
,
resistance
resistance
Charge
on
Permittivity
$0
Thermal
v*
Total
of
built-in
voltage
L
Inductance
Cp
FET
frequency
feedback
gate
function
frequency
on
material
potential
across
concentration
Angular
FET
semiconductor
equilibrium
(d
resonance
electron
reverse
Donor
voltage
subharmonic
an
e
Cg(vi)
p u mp i n g
the
n-side
junction
■•
2rF
capacitance
to
source
capacitance
as
a
o,f v o l t a g e
g^j
FET d r a i n
to
source
g ra
FET
Vj,V2
Instantaneous
S? ,
conductance
transconductance'
1 and
junction
voltages
on
varactors
2
?
v o i » v 02
Instantaneous
*
terminals
voltages
at
FET' s o u t p u t
M d » *■2 2
Instantaneous
current
through
the
feedback
current
through
the
load
I4
*
capacitance
i^
Instantaneous
*
d(
/
)/dt
VQ^»VR
resistor
"
Derivative
with
Instantaneous
respect
to
t i me
voltage
at
the
voltage
at
half
t
fundamental
frequency
v,,v0
- Ihstantaneoui
fundamental
the
frequency
i
‘
i wi —
V
— ii
I .
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 prohibited w ith o u t p e r m is s io n .
V(;g
Gate
Vjjg
Drain
v Tg
* o + VGS
(°)
Derivative
( ®° )
Second
■
Equals
wp
#
to
source
to
source
d.c.
d.c.
voltage
'
with
respect
derivative
by
voltage
to
w.r.t.
normalized
time
normalized
time t
definition
^
l / / L C g ( Vg s ) '
00p
1 / / LCp
Qp
1 / u pCg ( VgS ) Rs
P
> / u R
Y
w CF I p
X
Normalized
instantaneous
pumpi ng
voltage
S, Y
Normalized
instantaneous
voltage
at
the
voltage
at
the
subharmonic
%
frequency
»
Q, R
i
Normalized
fundamental
t
-
Normalized
instantaneous
frequency
time
■ cop t
“1
2gL + g G
“2
Sm
a3
go
a’
± e J ( VT-+'J' ) „ [ 2 4 ]
6
Phase
angle
of^X(o)
Phase
angle
of
Q(o)
’ Phase
angle
of
S(o)
$
12 ^ ]
f.
S
Siemens
..•.^skarSa
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 .
SYMBOL
-d e t e c t o r
phase
41 Q
Built
c g s <Ygs )
Cate
Vg S
Total
in
to
potential
source
forward
of
junction
varactor
voltage
capacitance
capacitance
between
^
th e ) gate
and
the
and
source
Total
Vj g
drain
to
Total forward
*
source
Vj
Threshold
X
"
Channel
ct
' v
source
voltage
length
modulation
tangent
Junction
P ,
Trans conductance
Rl
Load
V^.
Drain
v
. v 3c
Sine
zero
bias
small
sigpal
voltages
signals
resistance
DC y / o l t a g e
and
Cosine
^
voltage
Normalized
direct
X,Y
Normalized
sinusoidal
Z8 , Z C
Normalized
sine
,
at
parameter
sinusoidal
current
parameter
capacitance
X0 , Y ^ , Z 0
D
parameter
function
depletion
Instantaneous
Direct
drain
voltage
Hyperbolic
V 1 , 2 »3
between
,
C
V 1 ,2 ,3
current
Normalized
amplitude
current
and
drain
signals
small
cosine
D. C.
terms
signals
amplitude
terms
voltage
K
Normalized
threshold
voltage
P0
Normalized
transconduct'ance
a0
Normalized
hyperbolic
parameter
tangent
function
paramet e
9
Normalize(f j u n c t i o n
C0
depletion
capacitance
a tz e ro b ia s
gL
'
Normalized
(*)
Derivative
X',Y',Z'
Total
Xo
Normalized
output
with
normalized
transconductance
respect
to
‘
normalized
voltages
channel
length
modulation
p a r a me t e r
S(
w
)
Ritz
equation
Frequency
of
sinusoidal
signal
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 .
time
x
Integration
s ymbol
2P 0 / w CQ
^
x 0- z 0
Bo
C
*
.
,
•
,
3
Y0 - K
D-Z0
J
A2 0 ( I - B 0 ) ( B 0 - K ) 2
lo
1+X0D0 +2X0 (B 0 -K)
A Z
Normalized
D. C.
phase
difference
Phase
difference
variation
between
as
the
a function
t wo
input
/
Square
root
Abs o l u t e
d(
function
va 1 u e
Differential
)/dXc
v
with
respect
to
'
•
X0
K
f ak
____ -
• .
’
.
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 .
of
signals
«
t
LIST
OF
TABLED
Page
2.1
Material
for
use
as
thin
25
film 'res 'is to rs
J.
VI
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 .
CHAPTER
1
I NTRODUCTI ON
1.1
General
The
in
the
aided
growth
early
of
1970’ s
design
( CAD)
high
quality .gallium
together
has
led
arsenide
developments
to
the'
realistic
in
( Ga As )
computer
possibility
r
producing
monolithic
microwave
integrated
*
circuits.
of
*
Th e
o
birth
of'
Circpit.s
the
analogue
( MMI Cs )
Monolithic
industry
A
the
first
Mi cr owave”
wa s . l a u n c h e d
Jby t h e
successful
GaAs
MMIC
in
197 6
m o n o l i t h i c is used to
o
,
whe r e i n
all
a c t i v e and p a s s i v e c i r c u i t
4
ents
and
interconnections
are
f or med
term
surface
of
such^as-
epitaxy,
a
diffusion*
*’
used
realization
of
'
Th e
are
Integrated
to
s e m i - i n s u l a t ing
or
a
combination
a microwave
low
2.
I mproved
3.
Smal l
of
*
s
app.roach
elements
or
into
on
componf
\ ■
to
tlie
Deposition
schemes
or
s p u t t e r i n g ’,
these
an
processes
evaporation,
and
others
MMI Cs .
promising
1.
identify
substrate.
implantation,
fabricate
Some
fabricating
ion
[J].
attributes
monolithic
for.
circuit
designing
include:
co8t
reliability
size
and
and
r e p r od uc i b ilTTE y
weight
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 .
and
-
2
v *
-
•
*
4 A.
Mu 1 1 1 o c t a v e ^ ( b r o a d - b a n d ) p e r f o r m a n c e
5.
Circuit
' design
performance
*
ts
•
The
system^
on
flexibility
a
ch ip '.
cost
attribute
applications
based
of
on
a
MMICs
la 'rg e
'
of
are
considered.
space-borne.'
future,
which
require
Improved
from
MMI Cs ,
less
fact
least
critical
that'
within
fewer
size
and
and
numbe r
of
example
planned
and
is
wh e n
identical
the
for.
design
the
near
v
l ow c o s t
reliable,
reproducibility
w -ir e
the
and
realized
(2],
reliability
the
at
radars
lightweight
modul es
is
XI
A current
phased-array
transmit-receive
part
.multifunction
.
J.ow
components
and
bonding
chip
itself,
locations
at
is
and
/the
derives
in
eliminated
itr
is
relegated
to
p e r i p h e r y ’ of
the
chip.
Smal l
weight,
■ are
. , approach.
circuits
the
per
cost
qf
sizfe
wafer
is
processing
♦
intrinsic
Smal l
processing
volume,
that
processing
cost
per
batch
substrate*.
the
their
properties
allows
of
and
cost
the
chip
of
of
the
processing
Since
the
fabrication
entire
is
corollary,
wafer,
proportional
it
monolithic
of
hundreds
essence
is
of
of
batch
t
d e t e r m i n e d by
follows
to
light
the
that
area
the
of
the
of
the
V
chip.
*
*
The
^ undes i red
of
circuits
■?
1
reduction
parasitics
of
which
employing
bonds
limit
packaged
the
eliminates
broad-band
discrete
ma ny
if
performance
devices.
»
\ » \
>- „
«
wire
*
'
\
I
k,
•
’
!
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 .
-
The’
sm al'l
approach
allows
circuits
range
an
from
level,
for
80
GHz
or
realized
not
of
to
a receiver
to
a
level.
higher
front
for
monolithic
complexity
such
as
"functional
frequencies
been
These
end,
exceeding
have
the
a —c h i p
degree
bandwidths
woul d
on
amplifier,
been
with
f u n c t i o n s ' which
lowest
example
have
[3],
s i s e . ir ^ f r i ns i,c
integration
the
mixer,
MMICs
of
circuit
oscillator,
block"
circuit
30 GHz
possible
in
excess
for
circuit
using
conven-
c
tipnal
hybrid
which
have
circuits
phase
technologies.
been
designed
includes
.both
fehifters,
circuits
ma ny
0
poor
and
range
fabricated
s ma 1 1 - s i ’gn a 1
mixers,
of
analogue
as
and
^witches
GaAs
power
and
of
used.
these
basic
However,
applications,
integrated
amplifiers,
mu I t i - f u n c t i o n
MMIC
components
MMIC o s c i l l a t o r s . a r e
primarily
because
of
a r^>
The
noise
controlled
by
the
characteristics
resonant
of
. loop,
their
an
w hic'h
a
passive
factor
"Q"
i n d u c t o r ^ g a p a c i t,o r
of
a
r e s o n a n t . loop
for
to
the
oscillator's
phase-
noise
MMI Cs ,
The
inversely
[5].
proporr
ft
P a s s i v e MMIC
M
i
■
■,
1b
.
t
tional
are
com bination*.
Is
in
inherently
,
oscillator
~
generally
presently
being ^avoided
noise ^characteristics.
quality
circuits
[ A] ,
Many
being
The
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 .
.
4
» -•
components
sectional
are
very
areas
of
lossy
representation.
The Q
t
*
typically
le s s - t h a n 20,
i
an
MMIC
oscillator
oscillator
. noise
. '
to
reducing
and
this
and
phase*,
me a n s
cross
In
their
resonator
generally
Is
unacceptable
clean
of
phase
These
dividers,
not
locking
is
are
s hown
noisy
viable
locking
and/
frequency
thoroughly
in
oscillators,
filters
the
a
as
phase
mixerb,
only
reviewed
■.
phase locking
.loop
been
characteristics
different
of
components,
have
noise
source
composed
these
detector
for
reference
are
Of
>
the
.
configurations
frequency
detectors.
>
is
improving
solutions ,
configurations
\
by
Four
a
multipliers,
used
oscillator
•
for
MMIC o s c i l l a t o r s
thesis.
small
0
*
this
the^
■
applications,
A technique
i
'
lines
<
for
of
of
»
transmission
of
*
r
a-
because
t
the
-
phase
divider
investigated
in
MMIC f o r m .
#
This
phase
thesis
detector
presents
MMIC c o n f i g u r a t i o n s
for
a
frequency
h a l v e r 1.
Both
of
these
*
circuits
are
theoretically
analyzed
in
great
detail
and
<*
#
their operating crite ria revealed.
c
. An
tion
the
case
Ritz
of
a.nd
novel
approximate^ a l g e b r a i c
a nonlinear
steady-state
Method.
operation,
These
»
based
used
halver.
the
frequency
detector;
the
analysis
are
theoretical
subsets
on
e q u a t i o n , ‘ is
of
t wo
circuits
v
method,
differential
o f 1 the* phase
nonlinear
a
of
is
techniques
the
:
1
.,
for
'
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 .
solu­
obtain
In
the
on
the
analyzing
more' fam iliar
t
* '
to
based
analys i s .
.
the
Fourier
*
The
novel
c \ r cu i t s * f o r
the
frequency
halver
and
the
*
.
phase
detector
performance
are
is
designed,
then
fabricated
evaluated
and
and
measured.
compared
with
Their
theoretical
*
predictions.
* ' *
a
•
•'
i
-
^
''
1
"
1.2
Thesis
Objectives
Th e
following
►
>s '
.
I
research
reported
in
this
thesis
rela-tes
to
the
t
investigations:
*
a)
-
\
A search
a
for
practical
m o n o l i t h i ’c
*
component
microwave
implementations
phase
locked
<PLO) .
for
oscillator
‘
,
.
>
b)
The
development
of
the
theoretical
understanding
*
a
frequency,
halver
for
a
4
9
,
c^>" Th e
development
phas-e
of
detector
a
for
of
_
PLO.
/
.
a
*
'
#
theoretical
understanding
of
a
a PLO.
5
d)
Th e
experimental
of
a
monolithic
halver
c
1 ^ Thesis
design,
and
phase
k
development
microwave
and
evaluation
Integrated,
frequency
detector.
Organization
*
1
r •s
)
Chapter
-i, . . . . . .
• advantages
c
a
of
concept
practical
aa n d
1 is
the
introductory
monolithic
aicroyave
c h a p t e %r
'
describing
me d i u m
together
)
the
with
*
of
of
the
dilemma
phase
solution,
f requencyt halv e r
4
an
■
*
di-8 c u s s i o n
Th e
mi — - 1 - ■
of
locking
and
is
the
developing
is
MMIC
Introduced
necessity
for
a
-as
oscillators.
a
possible
phase
detector
explained.-
\
*
f .
'
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 .
'
-
Chapter
2 describes'
m icrow ave
Integrated
is
for
given
j
-
the
circuits
methods
are
by
which
fabricated.
implementation
of. t h e
A description
and
passive
a
employed
an
monolithic
active
*•
components
and
the
6
used
is
in
laid
a
out
underlying
MMIC
in
design.
^he
a systematic
consideration
of
design
methodology
step-by-step
wafer
yield
procedure,
is
introduc­
ed.
In
phase
Chapter
locked
/
of
the
length.
Also
detector
and
state
In
>
the
Is
the
together
with
a
an
system
components.
The
*
individual
oscillator
building
scenarios
Inderstood
frequency
4,
the
halver
frequency
Investigated.
Chapter
is
5,
- An
block
are
are
of
the
discussed
components,
the
singled
*
phase
the
halver
is
analytical
described
at
phase
out
for
in
jnodelled
solution
and,
for
the
detail.
detector
is
modelled
and
,
'
investigated.
An
.
.
*
the
required
least
response
^theoretically
’
the
Chapter
theofcetically
/
presented
of
investigation.
*In
f
*
each
phase-locked
,
configurations
*-
of
different
steady
are
r,
requirements
further
possible
Borne
oscillator
identification
*
3,
individual characteristics
presented.
phase
O p t i m i z a t i o n . of
detector
*
greng i t i y f t y
*
'
4
analytical
of
the
is
the
bias
solution
phase
based
detector's
voltages'to
derived*.
*
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 .
on
FETs
maximize
Chapter
6
V
carried
describes
the
laboratory
measurements
f
put
to
confirm
the.theory
developed
for
the
frequen-
fv»
cy
halver
and
measurements
dr awn
are
regarding
In
presented
the
compared
the
Chap t e r
and
phase
detector.
with
accuracy
7,
a
of
theory
the
s u mma r y
recommendations
results
of
the
and
conclusions
are
models.
of
for
The
the
^
achievements
further
research
given.
- ^
- ■
- - *
■-
*■
■■
___
*------------------------------------
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 .
is
are
CHAPTER 2
Ga As r MONOLI THI C MICROWAVE INTEGRATED
2.1
Processing
C I RCUI T S
Techniques
/
Monolithic
complete
In
electrical
addition
circuits
to
The
advantage
of
requires
active
chip
size
mi ze
the
methods
minimized
number
of
especially
chip
chip6
whereas
device?
18
circuit
f a b r i c a t i o n ' on
2.1.1
as
relatively
small.
a GaAs
Integrated‘ Clrcult
chip.
diodes),
these
area
The
in
The
tend
to
occupied
next
and
realize
and
required
possible
elements,
the
single
(and
dictate
slice.
are
inductors,
fabricate
much^as
a
( MMI C s )
reproducibility,
to
on
a
and
MMI Cs
considerations
inductive
area,
( FETs
fabricate
exist
on
capacitors,
ru g g e 'd n e ss ,
Economic
be
devices
t-o
circuits
fabricated
include
ability
size,
that
components.
the
the
integrated
circuits
generally
resistors.
ents,
microwave
the
cost)
passive
that
the
MMIC
order
tV maxi­
passive
compon­
utilize
mos t
by
section
the
of
active
deals
with
substrate.
Technology
r
r
A
cornerstone
availability
of
a
of
highly
the
monolithic
reproducible
approach
device
^is
the
technology.
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 -
■
y
<
This
in
turn
starting
is
relatef,
material,
in
part,
especially
to
the
the
active
control
of
the
(semiconducting)
layer.
Two
an
active
general
layer
implantation
In
on
Ga As
the
substrate.
available*
substrates;
epitaxy
layer
namely,
"buffer"
deposited
an
layer
from
approach,
is
Usually
impurities
are
the
ion
directly
into
strate.
Ion
the
a
is
used
to
single
crystal
es e n l - l n s u l a t l n g GaAs
a
intervening
high-resistivity
screen
out
during
larger
surface
implantation
in
[S j\
epi­
diffusion
the
of
active “ layer
different
uniformity
implantation.
does
with
of
the
device
higher
not
layers
in
parts
a
have
are
doping
the
formation
the
Wi t h
of
imp'lanted
GaAs
as
sub­
quality
control
or
implanta­
possible
level
implantation,
of
are
crystal
implantation.
with
is
at oms
s e m i - i n s u 1 a t i ng
requires
unitorm
This
dopant
a
conducting
Furthermore,
possible
profiles
mo r e
of
Epitaxy
uniform
area;
thickness.
with
the
implantation,
associated
mo r e
higher
ion
doped
on
substrate
substrate.
flexibility
is
and
r
Using
tion,
forming
epitaxy
growt h .
of
for
[6,7J.
semiconducting
taxial
techniques
well
over
as
a
in
selective
doping
different
doping
wafer.
characteristics
In
is
k
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 .
addition
a
achievable
^
The
is>
also
processing
applicable
elements.
The
^associated
fpr
the
Many'
to • the
high
with
FET
pircuit
technology
there
arp
process
technology
incorporating
layer
on
to
a
s hows
the
techniques
a
FET,
the
d e p /a s'itio n
adequate
implants,
Shallow
Se
implant
diodes.
the
[Figure
contact
The
already
discrete
between
FETs,
the
epitaxial
steps
t wo
layers
ion-inplanted
of
the
a
device
as
and
implantation
(silicon
channels
is
then
deeper
areas,
wafer
substrate.
the
n+
or
steps
IC
is
nitride)
The
first
areas
which
of
FETs.
followed
implant
Schottky
are
the
switching
IC
circuit
of
GaAs
a
Ga As
and
2.1(a)]
one
planar
and
S i 3 ilc,
GaAs
for
a
shifter
of
such
photoresist
device
of
have
planar
fabricate
seml-insulating
dose
the
the
level
l ow
for
definition
than
ICs
-where
process
Fabrication
the
GaAs
resemblance
of
to
a
require
2.1(b)]
mo r e
circuit
follow.
defines
of
is
fabrication
particularly
stage
exposure
the
marked
photoresist
switching
:■
by
passive
dimensional
developing
in
will
diode.
initiated
fabrication
-
discussion
2.1
possible
Schottky
be
brief
Figure
many
to
processes
A
FET
elements.
tends
used.
of
photolithography
laboratories
fabrication
for
monolithic
degree
c o n s i d e r a b l e >e x p e r i e n c e
thus
used
by
..........................................................................................................
a
[F ig u re
barrier
followed
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
by
-
11-
Ptanar GaAs IC fabrication step s
Insulator d ep osition and masking lor N- implant
InsuWfor/ Photo resist
i l l l l l l t l l r — -*ilr-i I L g
—
S c m. m m W iw g Q * * > -
•S’
N* Im plant
—
(b)
S, 1 GaAs —w—
—*
)
/
E n ca p su la tio n and anneal
, Multi-tayer dieffecinc
— y______S l G aA s
(c)
O h m ic c o n t a c t m e ta llisa tio n
A u G e /P t c o n t a c t / Insu lator
S I G aA s
S c h o t t k y barrier a n d in te r c o n n e c t m e ta llis a tio n
t i o .,*
. S c h o ttk y barrier
Ji/P t/A u g a te r
^ m terco n n ect
S e c o n d la yer m e ta llis a tio n
**-----
_ea
— la y * l
Insulator
n <5I»
L ^ r -'—
~T
(f)
, C u t a w a y v ie w of a p la n a r G aA s circuit fab rica tio n with tw o
l o c a li s e d im p l a n ts .
P la n a r G aA s l C
^
First-second
S e co nd-le vel
S e c o n d - le v e l i n t e r c o n n e c t via
✓i n t e r c o n n e c t
in s u l a to r
S u b strate
in s u la to r
O h m ic
contact
F ig .
by
L
\ F i r s t level
in te r c o n n e c t m e ta l
' P l a n a r GaAs C i r c u i t F a b r i c a t i o n P r o c e s s a d o p t e d
Rockwell.
This u ses Ion I n p l a n t a t i o n D i r e c t l y i n t o
Q u a l i f i e d S £ i » i - I n s u l a t i n g S u b s t r a t e M a t e r i a l . 19]
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 c o py rig ht o w n er. F u r th e r re p r o d u c tio n prohibited w ithout p e r m is s io n .
t*
'
i
>
12-
-
encapsulation
of
annealing.
This
implanted
the
areas
D e v i c e , ohmi c
photoresist
'ih e
contacts
After
with
annealing,
into
and
r*
slice
lift-off
S i 3 Ni+
defined
the
ohmi c
high ,te io p e ra tu re
the
active
shallow
channel
next
technique
alloying
and
converts
FET
are
»
n-Se-
layers
with
a
[9].
standard
(Figure
2.1(d)].
contact
areas,
a
photo-
C
resist
operation
barrier
uses
layer
dielectric
as
again
metallization
Ti-Pt-Au
first
is
for
the
insulation
is
for
and
second
view
as
well
[Fig.
pn
to
layer
Schottky
Rockwell
as
for
the
2 . 1 ( e.) ] .
the
entire
A
wafer,
interconnectibn
capacitors.
of
the
diodes.
barriers
deposited
the
circuit
cross-sectional
the
FETs
define
interconnections
then
for
to
Schottky
circuft
layer
dielectric
for
performed
Fig.
t h e , \ goip 1 e t e
2.1(g)
process
and
shows
where
a
the
i
interconnects
discusses
2.1.2
the
FET
Th e
conductor
gained
can
Fabrication
t wo
mos t
etched-channe1
exploit
clearly
different
Field
the
be
either
FET
fabrication
used
for
following
section
techniques.
Ga As
Transistor)
popularity
technologies
optical
The
Techniques
techniques
Effect
seen.
or
are
the
[10,11].
electron
MESFET
(Metal
fabrication
which
self-aligned
Both
beam
Semi­
have
and
the
techniques
can
lithography
to
■/
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 .
define
the
length.
gate
stripe,
Electron
lengths
^
of
depending
on
the
beam . 1 i t h o g r a p h y
\
less
transistor's
Is
generally
Self-Aligne,d
fdx
the
t wo'
*
Gate
»
gate
used
t\an
.5 u b .
A d i s c u s s i o n of
/
J
'*
FET f a b r i c a t i o n t e c h n i q u e s i s g i v e n b e l o w .
y/
/
'
»
different
j ■&
!
gate
’
-
Fig.
2.2
self-aligned
t i on
mesa
until
the
metal
is
Fig.
s hows
gate
is
.the
technology.
defined
by
2.2(a).
In
the
over
and
first
a wa y
substrate
evaporated
Source
processing
etching
semi-insulating
then
basic
the
drain
steps
step,
the
of
reached.
active
area
the
t»he I s o l a -
active
is
areas -are
»
n-layer
Th e
as
gate
s h o wn
defined
in
in
photo«
resist,
and
t he r
exposed
gate
metal
is
r e mo v e d
by
etching.
o
Over-etching
is
used
Fig.
to
allow
2.2(b),
drain,
and
gate
evaporated
the
gate
metal
and
is
to
[Fi^.
strij|
off”
left
situated
in
Fig.
2.2(d).
-L_________ —________ t_____________
the
2.2(c)].
is
is
undercut
source.
covered
conveniently
"floating
to
necessary
between
resist
space
o h mi c
contact
Th e
resist
which
with
this
gold.
source
by
ohm| c
dissolving
Thus
and
the
drain
as
s h o wn
in
gate
to
between
Gol d
r e move d
the
the
metal
Is
is
then
protecting
m e t a l , - but
a wa y
the
remaining
contacts,
=----------------------
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 .
resist
thin
as
the
gate
s h o wn
-
14-
s
Recessed
In
Channel
thistechnology,
defined
by
thicker
layer
the
is
grown,
by
etching.
required
region
not
is
first,
shown
defined
inp h o t o r e s i s t .
until
specific
drain
excess
a
me t a*4
technique^
[Fig.
.This
Gate
The
Improve
recess
thereby
next
the
can
sect i
in
the
basic
Fig.
and
recessed
short
channel
small -signal
decreas.e
the
removed
works
the
.25
source
the
and
in
the
and
and
the
"float-off"
wel l ,
with
^exposure
pm
be
Ga As
is
Ga As
source
resist
can
are
gate
the
the
equally
both
tech­
produced.
dan
significantly
FET
[12].
resistance
figure
the
and
using
the
contacts
evaporated
technology
performance
noise
discusses
beam
as
l a y e r , . when
betwj^n
by
the
as
is etched
then
under
a
higher.toler-
drain
2.3(a),
is
electron^
as
an$
measured
method
the
is
.layer,
region
epitaxial
channel
is
2.3(c)]
lengths
improve
removes
Gate m e t a l
photolithographic
niques.
This
Source
A
current
contacts.
channel
for
thickness
n-type epitaxial
the
etched.
as
active channel
the
and
thickness
deposited
the
h e i g h t of
defined
channel
Technology
of
the
associated
fabrication
of
capa c I t o 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 .
Such
device
gain.
the
a
and
The
MMIC
-
15-
^
Photoresist
P h o to re s is t
'
(_J3ate metal
■Gate metal
1 ~ 2 '
Y////7777777/y///// n layer
S u b s tr a te
(a)
O h m ic metal
P h o to re sist
S o u rc e
W /7 7 7 ///7Z Z Z " 7Z Z A
G a te
Dram
E z z z lT z k z z z z z L
(d> .
(c) I
Fig.
2.2.
Processing
Te
Steps
of
Self-Aligned
Gate
c h n o l o g y . [ A)
P h o to re sist
O h m ic m etal
nlayer
S u b s tr a te
<b)
(a)
G a te m etal
S o u rc e
Dram
G ate
(c)
Fig.
2.3.
Processing
Steps
of
E t c h e d - C h a n he 1 T e c h n o l o g y . (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 .
« .
i
2.1.3
Capacitor
16-
Technology
4
Capacitors
ed
to
as'
fabricated
rac^Mlithic
blocking,
*
as
part
capacitors.
bypassing,
or
-of
They
tuning
an
are
MMIC a r e
used
purposes.
referr­
in^ MMI Cs
There
for
are
four
*
basic
MMIC c a p a c i t o r
transmission
formed
k
line„
as
an
configurations.
[Fig.
2.4(a)].;
interdigital
meta1- i n s u l a f o r - m e t a 1
( MI M)
These
are:
coupled
capacitor
in
a
l i n e s ,'^ V s u a lly
[Fig.
capacitor
a s ti i sb
[Fig.
2.4(b)];
2.4(c)]
a
and
a
is
a
Schottkydiode.
Schottky
function
of
voltage.
diodes
diode
The
unsuitably
for
on
area,
GaAs
voltage
use
in
Ga As
have
analog
profile,
ma k e s
a
capacitor
quarter
or
a
an
wavelength
stub,
less
stubs
are
area
\
in
at
than
a
mic.rostrip
>
*
inductor.
in
quarter
distributed
lower
The
applied
structures
■
open
is
line
stub,
' less
capacitive.
and
is
require
can
A
act
than
as
a
shorted
inductive.
Such
substantial
chip
frequencies.
mo r e
the,
popular
digital
and
types
MMIC c a p a c i t o r s
of
An
wavelength,
elements
and
these
transmission
length,
that
MMIC a p p l i c a t i o n s .
.
A stub
capacitance
doping
dependence
most
a
capacitors
meta 1 - i n s u 1a t o r - r a e t a 1
will
now
be
are
the
types.
described
inter-
These
in
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 prohibited w ith o u t p e r m is s io n .
detail.
t wo
Interdigital
Capacitors
*
■***
.*
Th e
from
capacitance
coupling
substrate
not
chip
a
tance,
of
area
an§
interdigital
adjacent
Th e y
1 pF o r
pure, series
are
less.
involve
and
circuit
usedfor
useful
Larger
aided
by
the
principally
for
but
effects.
have
capacitance.
mofiling
Th e y
are
Fig.
2.5
s h o ws
interdigital
an
struc­
x
Th e
capacitance,
Th e
capacitance
C
signif­
associated induc­
[13,14],
tance.
*
values' require
distributed
shunt
'
capacitors arises
conductors,
c a p s c i t a/n c e ,
resistance,
equivalent
tures
between
dielectric.
capacitances
icant
of
-
(
N
F-1
)
C,
is
C can
the
be
i n t e r e l e c t rode
calculated
using
capaci­
[13]
2
C g L
.
1
a
where
Np
is
the*
and
is
the
static
Cg
unloaded
35
■'*'
to
current
n u mb e r of
gap
these
capacitors
at
12-14
GHz .
50,
crowding
near
greater
consideration,
for
the
design
the
Q
Th e
and
is
is
the" f i n g e r
per
unit
typically
losses
edges
values.
The
of
the
prevent
major
capacitors,
spacing
in
the
i s'
length
length.
associated
conductor
interdigital
size
L
capacitance
Q of
significantly
achieve
fingers,
range
with
The
of
the
obtaining
fabrication
to
accurately
conductors.
( .
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 .
-
18
STUBn^
\
11
MICROSTRIP LINE
b)
*)
DIELECTRIC
Fig. 2.4.
T h r e e I m p l e m e n t a t i o n s of c a p a c i t o r s o n , M M l C s : ( a )
a Btub in n l c r o s t r l p t r a n s m i s s i o n l i n e ;
( b ) an i n t e r d i g i t a t ed c a p a c i t o r ;
( c ) a m e t a l - l n s u 1 a t o r - m e t a l ( MI M) c a p a c i t o r .
r\
-V N /— rrrv —
| ^
c
/v
F ig. 2.5.
capacitor.
r
X
E q u i v a l e n t c i r c u i t of an
The s h u n t c a p a c i t a n c e -is
plane.
lnterdigltated series
to the backside ground
-
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
-
19i
Metal-Insulator-Metal
•4
n
Capacitors
*
*
/
The
MIM
*
mos t
popular
configuration
^
type
s hown
capacitor
is
composed
dielectric
material.
of
in
of
The
capacitor
Fig.
t wo
for
2.4(c).
metal
This
plates
dielectric
is
MMI Cs
is
the
thin-film
separated
typically
0.1
by
to
A
0.4
inn t h i c k .
Because
capacitors,
by
the
these
their
standard
<
C0
to
the
first
parallel
plate
order,
given
eA/d
is
is
the
-
p a r a l l e l 1 plate
‘
capacitance,
2.2
t
is
the
and
typical
the
thicknesses,
of
50
capacitors,
*
capacitance,
are
p e ' r m i t t i v i t y , A i s t h e are.a o f ' t h e t o p p l a t e , and d i s
‘
’
d i s t a n c e between p l a t e s .
F o r c o mmo n l y u s e d d i e l e c t r i c s
range
v .
structures
expression:
C0
whdre
MIM
tive,
C,
Rp ,
Cp) ' ,
as
2.6.
to
also
capacitance
’
600
have
2
^pF/ mm ,
-in
Lp *
the
typically
However,
ot h e r ^ p T r V s T ? i c
inductive,
illustrated
is
and
these
components
shunt
equivalent
in
series
(reals-,
capacitive,
_circuit
in
Fig.
[14],
can
be
(“ 1
pF)
f^ioxide.
is
GHz ,
even
the
less,
Q of
a
while
large
the
MIM c a p a c i t o r
Q of
a small
( ’■10
Mill
pF)
capacitor-
can
or
the
it
10
10 o r
losses
so
2
Op ,
At
fo r
p F / mm
this
the
e x c e e d 100.
Th e l o s s e s
are e ith e r
dielectric
/
*
conductive - losses
(14).
Th e m o s t common c h o i c e
c a p a c i t o r
These
n a t u r a l
'bv d i e l e c t r i c
are
to
often
u se
used
them
is
s i l i c o n
in
the
for
n i t r i d e
or
encapsulation
c a p a c it o r
s i l i c o n
process,
d lele.ct r ica
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'
Ib o .
-
The ‘ y i e l - d
Important
however
the
issue.
pinhole
entire
chip. •
per
large
capacitor
• 9 5 10
•
6 OX,
-s
working
Some
one
reasonably
\
of
which
- due
be
ruins ^ n e
9 5X,
to
of
if
MMIC
the
s l i c e is
Hence
an
capacitors,
also
which
ruins
has
ten
fabrication yield
t h i s you Id p r od u c e
capacitors.
a
capacitor,
complex
capacitors-,
on
a
use
make-liberal
Considera
'-
mil s t
capacitors
*
HMICs
MIM
is
20-
a
chipy i e l d
the
capacitor
of
yield
^
ve ry ' h i g h .
,
Another
passive
MMIC i n d u c t o r ,
important
yhich
Is
described
circuit
in
the
component
next
is
the
section.
J
r-W
L f,
Fig.
2.6.
2.1.4
E q u i v a l e n t c i r c u i t of a me t a 1 - i n s u 1a t o r - m e t a 1
( MI M) s e r i e s c a p a c i t o r
[14].
Inductor
Technology
r
A
Inductors
tftey
function
l umped
treated
MMI Cs
are
^ —i
or
part
necessary
of
distributed
as
are
as
are
tuned
[15],
transmission
usually
achieved
usjing
between
short
elements
circuits.
S^The
lines.
0.5
in
and
They
are
distributed^
Inductor
2 0 n H.
transmission
MMl Cs ,
where
either
forms
values,
Smaller
/ines.
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 .
on
are
Ga As '
inductors
a
r
-
/
L u mp e d
Fig*
tors.
Inductors
2.7
.These
Inductance
Illustrates
are
le n 'g th s
between
interactions.
■^described
in
21-
several
of
metal
segments ,
These
several
types
lines
because
monolithic
references
of
lunped
induc­
have
nutual
that
of
electromagnetic
lum ped
[4,16,17],
inductors
Their
are
small
site
%
ma k e s
have
modelling
been
treated
directly
been
developed
in
reference
reference
[17],
on
used
elevated
difficult,
the
[16],
the
and
these
substrate,
i n-
s o me
Rectangular
Although
[18],
above
(16).
although
which
mos t
spiral
circular
inductors
an. a f r
bridge
of
the
s e mi - i r i f e u 1a t i n g
accurate
equations
Inductors
spiral
are
Inductors
usually
spiral
surface.
length
The
in
placed
inductor
inductor
are
Q' s
has
is
of
<4
l umped
inductors
inductors
having
tend
smaller
to
be
near
^50
Q's.
at
10
GHz ,
j
m
X
Fig.
2.7.
L u mp e d
inductor
J
configurations
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 .
longer
-
Distributed
As
cult
to
form
the
Ga As
noted
model
aided
account
For
almost
always
lines
transmission
programs,
they
Join
purposes',
microstrip
Th e
have
been
used
transmission
on
the
essentially
as
lines
in
do¥s^ not
mutual
couplings,
these
transmission
the
a l t h o<*-g-h
coplanar
Both
impedance
substrate
disadvantages.
atrip
pure
the
because
coplanar
most
ance
of
width
the
lines
First,
of
are
the
mo d e
popular
✓
the
to
line
the
is
choice
is
and
s o me
transmission
magnetic, ^nd
lateral
difficult
for
in
radiative
mor e
ground
to
space)
configura-
the
dielectric
losses.
The
t wo
micro-
Second,
lines.
thickness,
are
have
• Microstrip
ratio
lines
the
dielectric
than
the
factors
strip,
planes.
obtain.
2^.8 .
N
lines
by
secondary
Fig.
coplanar
#
transmission
determined
substrate
substrate,
microwave
the
occupy
are
transmission
/
s hown
m a t e r i a l . 1 Coplanan
they
and
lines
depends
on
the
dimensions
of
the . c w t r e
j •
spacing
to
the
lateral
ground
planes,
and
the
of
computer
accurately
tion
constant
to
o th r ^ metals.
[19,20].
line
d iffi­
preferable
,
lines
are
fabricated
function
capacitance,
where
MMIC
as
inductors
generally"
they
modelling
associated
discontinuities,
is
transmission
treatment
( CAD)
for
as
It
'Although
their
design
lumped-element
accurately.
Inductors
inductors,
-
Inductors
above,
substrate.
22-
of
The
the
is
imped-
conductor
constant
of
Losses
in
[21].
dielectric,
a
conductive,
transmission
itihi
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 .
lines
must
be
sufficiently
significant
be
at
*
the
ohmi c
least
applies
for
cross
t wo
RF
skin
carry
required
me a n s
depths
thick
U
mus t
If
be
DC
to
This,
are
of
should
course.
present,
support
component
next.
without
the .metal
[21].
currents
MMIC
current,
that
sufficient
passive
discussed
the
This
currents.
Another
is
to
losses.
section
density.
which
thick
is
the
the
f
then
current
resistor,
___
ELECTRIC FIELD LWW
S l DiAi
GaAs
1
Fig.
2.8.
Two
basic
MMI Cs :
2.1.5
Resistor
and
MMI Cs
are
using
the
GaAs
The
18
approach
are
to
of . t r a n s m i s s i o n
coplanar;
U 6 e d
in
(b)
lines
used
on
microstrip.
MMICs
isolation,
dividers
divided
Ga As
-* s l i c e
(a)
feedback,
voltage
types
Technology
Resistors
including
MET AL
6)
in’ bia6
here
into
material,
and
for
several
s e 1f - b i a s i n g ,
networks
t wo
terminations,
[22].
classes:
thin-film
purposes,
Resistors
those
on
fabricated
resistors.
Re s 1 s t o r s
' V"' simplest
use
requires
the
no
way
to
fabricate
resistivity
extra
of
process
a
the
•«*
resistor
GaAs
steps,
on
itself.
because
*
V
*
the
•f-
— __
at
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 .
Ga As
This
the
t wo
0
-
major
Isolation
steps,
already'
performed.
be
2.9,
The
resistance
GaAs
materia
contacts.
The
is
tance
for
a
resistance
tance
is
around
current
serious
large
to
.25
o h m- mm.
resistors
The
resistance
positive,
changes
over
Rc
and
Fig.
ohmi c
is
the
-
I
and
(ohms/
contacts,
contact
values
and
produce
W is
resis­
of
contact
sheet
resis­
problems;
temperature
is ^ the
coefficient
can
C i*i
thp
ohmic
Ga As
coefficient
temperature
of
t wo
potential
temperature
I______ »—C|»l _
*1
the
formation,
thus
the
Typical
te^per^fure
The
of
several
domain
Fig.
In
2. 3
ohms/square,
have
Gu n n
disadvantage.
and
45 0
shown
„
the
( o h m- mm) *.
mus t
[23]
*
resistance
and
contacts,
resistance
of
by
between
is
the
-
300
[24],
both
given
resistor,
saturation,
coefficient
is
distance
length
ohmi c
resistance
shee„t
the
unit
of
2R
+ -rr^
W
W
the
are
Ga As
the
of
resistor
t*
the
square),
L is
I
the
width
of
a
consists
resistance
S
Rs
formation
Such
and
R ■ R
where
and
24-
is
most
both
significant
[4],
_
/
2.9.
MMIC r e s i s t o r c o m p o s e d o f a c t i v e GaAs m a t e r i a u l
d e f i n e d by m e s a e t c h i n g . ; ( b ) d e f i n e d by i m p l a n t
i s o l a t i o n or s e l e c t i v e i m p l a n t a t i o n
[4].
(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 .
<
v
I
-
Thin-film
The
use
25-
Resistors
of
GaAs
material
It
is
therefore
materials
to
provide
as
resistance
has
several
P
disadvantages.
thin-f*ilm
course,
have
be
been
compatible
used
Re
th i* s
stors,
require
at
pattern
( Ga As
isolation
for
least
one
ohmi c
Having
purpose
contact
described
will
now
be
all
Materials
are
for
listed
GaAs
fable
re^^^t ors,
maski ng
level
fabricated
the
to
using
of
that
2.1.
generally
define
the
MMIC c o m p o n e n t s ,
the
existing
the
design
explained.
Us e
as
2. 1
Thin
Film
Resistors
Material
Resistivity
(p o h m - c m )
Cr
Ti
Ta
TaN
Ta2 N
Ni
NiCr
GaAs
Au Ge Ni
13 ( b u l k )
55-135 (43 b u l k )
18 0 - 2 2 0 ( 1 3 b u l k )
28 0
300
7 (bulk)
6 0-6 00
3 0 0 - 4 5 0 ohms / s q' .
2 ohms/ s q .
(alloyed)
in
must,
masks).
TABLE
Material
other
processing.
additional
be
consider
These
than
can
to
resistance.
GaAs
other
resistors
and
methodxilogy
with
useful
[A]
TCK
( p p m)
3000
25 0 0
- 1 0 0 t o +5 00
-18 0 to -300
-5 0 to - 110
2 00
3 0 0 0 - 3 2 00
/
NOTE:
The e x a c t
value
that
is
obtained
is
dependent
on
deposition parameters.
In a l l c a s e s , t h i n - f i l m . r e s i s t i v i t y
Therefore this data
w i l l be g r e a t e r t h a n b u l k r e s i s t i v i t y ,
s h o u l d be u s e d o n l y a s a g u i d e 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 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 .
2.2
MMIC D e s i g n
Methodology
4
The
not
yet
design
matured
established
finished
satisfy
of
to
the
methods
designs,
the
is
point
to
a
relatively
where
a
proceed
which
can
designer
from
be
new
art
and
may ^ u s e
well
specifications
successfully
has
fabricated
■L
to
to
req u i rements .
Every
specific
MMICs
new
designs
art
mus t h a v e
%
attempted.
are
its
foundations
In
monolithics
laid
before
y
these
found-
>
ation
blocks
capacitors,
designer
passive
are
the
passive
resistors
and
understands
'the
circuitry;
the
interconnects.
CAD
components,
a design
At
a
of
.inductors,
Until
a
representations
an
MMIC c a n
take
circuit
of
these
ma ny
itera­
tion's.
the
present,
development
( MMAC)
as
an
realization
processing
circuit
of
90
are
is
elements,
for
MMAC ( p r o c e a 8 i n g .
development
of
design
stage,
[22].
except
on
for
the
passive
and
before
MMACs
MMACs
for
the
circuit
fabrication
can
a
MMICs
to
This
ajr^d
of
the
batch
fabricate
all
which
allows
because
process
stepping
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 .
It
ti^ re
steps
stones
MMI Cs .
are
separate
active- devices.
turnaround,
as
circuit
monolithic
devices,
fabrication
serve
fully
mos t
active
involves
active
use
substrate.
fewer
procedure
miniature
developed
faster
critical
used
microwave
attempted
technology
allows
less
a
intermediate
wirebonded in pl ace
1
o p t i m i z a t i o n of t h e
al
wi dj e l y
in
the
-
For
design
the
is
me nt s
dictate
evolved.
step
device
is
is
the
witl
Using
a
types
usually
at
have
MIC u s e s .
device
When
MMI Cs ,
an
the
mo d e l
for
the
active
aided
design
The
of
the
be
t i m e as
accessible,
the
data
measured
then
used
to
require­
Th e
on
of
The
active
since
it
the
circuit.
active
devices
them
obtain
top­
used.
characterization.
separately
or
system
to
the
a
from
previous
mo d e l
fo^'the
or
p r o c e s s i; - £
plant
equivalent
devices.
device,
one
is
used
circuits
After
can
to
are
normally
obtaining
proceed
to
process
a
the
circuit
computer
phase.
CAD
circuit
step
the
topoV ogy
tolerance
success
elements
processing plant',
active
( CAD)
the
and
approach
and
rest
external
for
tion,
point ; is
i n ' a MMIC e n v i r o n m e n t .
available
the
circuit
the
is
S-pararaeters
In
be
s a me
data
a
active
device
extensive
This
starting
c irc u it topology
not
the
the
which
of
active
company's
normally
active
from
Th<e
the
fabricated
into
design,
specification,
ology
next
MMIC
27-
of
this
device
and
device
file
studies
step
to
allow
t o be
depends
circuit
data
are,
incorporated
analysis,
optimiza­
performed
on
the
circuit.
directly
on
the
accuracy
in
the
computer
completed,
the
mode l s
used
programs.
After
is
the
mask
cut-and-try ,
to
use
a
the
circuit
design
layout.
Since
or
on
based
computerized
mu c h
prior
maslc
is
of
the
layout
experi^ej^pe ,
layout
system.
it
is
There
next
step
process
is
convenient
mu s t
be
&
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
-
capability
the
CAD
steps
for
step,
interaction
and
in
28-
between
general
necessary*
o
Once t h e c i r c u i t
the
ma n y
layout
process
and
iterations
1between
the
are
coordinate
ma sk
data
layers
0
(external
layout
complete,
the
d e s c r i b i n g ' the
topologies
*
■
v
recorded
on m a g n e t i c
tape.
are
fabrication
service)
The
tape
is
then d e l i v e r e d .
sion
of
this data
generators.
is
to
Th e
actual
W '
is
being
following
*
u s (e d ,
by
photomasks
the
various
a
foundry
* If
step
photomasks
completed
of
numerical
the
is
magnetic'
the
conver­
automated
are
sent
to
pattern y
the
tech-
»S
ir o l ,o g y l a b o r a t o r y
where
they
—
are
inspected,
and
if
satis - (
*
factory,
are
used
in
the
next
tion.
namely,
circuit
usually
dc
s
~ Th e
wafer.
have
fabricated
The
passed
wafer
the
packaged
and
the
foundry,
often
of
In
before
the
a
test
delivered
practice,
(desired
•o
perfected,
is
probe
into
c h i p s and
are
mounted
on
quite
1
Thisc o n t r i b u t e s
to
the
probed
the
several
circuit
design,
design
fabrications,
is
that
for
RF
RF p e r f o r m a n c e
In
the
case
performed
iterations
performance
on
chips
carriers
customer.
o n l y dc m e a s u r e m e n t s a r e
»I 1
f r a c t i o n of t h e w a f e r c h i p s .
On c e , t h e
'wafers
diced
are
The c h i p s , w i t h s a t i s f a c t o r y
, are
delivery
circuits
is
dc
measurements.
are
step;
*’
fabrica­
are
of
before
necessary
obtained.
and t e s t i n g
processes
r '
the incremental cost
’s m a l l
to
a
compared
low
cost
to
per
to
the
chip,
produ«
additional
development
provided
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 .
cost.
that
the
-
develttpment
number
are
cost
of
for
yield.
2. 3
can
be
and
well-controlled
units,
used
29-
spread
fabrication
over
and
a
sufficiently
automated
testing,
large
techniques
resulting
im
high
t
Yield
Considerations
>»
Y i e 1 d - 1 i mi t i n g
r
generally
classified
Th e
significant
mos t
related
to
s i'm p i e
factors
as
dc
in
either
catastrophic
y i e 1d - 1 i m i t i n g
failure
\
processing
MMIC
of
or
factors
a-fe
parametric.
are
MIM c a p a c i t o r s
,
a nd
usually
GaAs
FETs
k
[23].
. These
failures
])art i c u l a r l y J
alignment
of
factors
the
insulating
between
the
the
to
which
and
first
minimization
and
of
processing-related
contact
limit
-and,
lithography,
of
metal
first-level
mental
deposited
achieve
a
high
t.o leran ce.
The
levels
MIM
capacitor
to
mask -
of t h e
theuse
the
6
thickness
dielectric
yield
smooth
and
of
of
the
layer
MIM
include
of
short
metal
dc
[26,27,28].
yield
iaclude
etching
edges
of
the-
circuits
second-level
periphery,
produce
electrode
of
of
yield
constant
capacitor
between
to- produce s h o r ts ,
c on t ac t the top
v
%
Control
dielectric
occurrence
second
overlap
MIM c a p a c i t o r
and
the
j naxi mi ze
first-level
the
case
thickness
layer,
Approaches
likely
both
related.
The
control
in
are
,
that
and
the
of
the
a
air
less
bridges
to
constant
of
capacitor.
and
is
di.electric
necessary
capacitors
to
In
a
order
to
specified
4
thickness
of
the
deposited
silicon
*
.
’
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
nitride
*• ♦
layer
can
tor,
ing
and
controlled
the
dielectric
constant
stoichiometry
'
result
and
FET
of
of
FETs
dc
are
processing
with
reduced
tion
for
yields,
2.4
to
broken
or
occur
(1986)
with
spacing
yields
DC,
is
Test
in
GaAs
integrated
reduce
the
circuits
has
as
previously
been
well
reported
configured
as
of
for
JB
R F ^ j u a n t i t i es
an
by
a
for
gate
a
of
length
for
fabrica-
but
mus t
measuring
next
be
of
high-
the
wafer
section.
System
wafer
and
prior
Eddison
as
incidence
FET
’
microwave
automatic
such
are
*
the
measuring
The
In
[29,30]. *
Equipment
effort
MMIC e v a l u a t i o n ,
Development
been
on
subsequent
conventional
^
to
total
gates.
production
the
*
ability
f a iVl u ”r e
circuits,
viable
*
perforin
dc
are7 adequate
»
A technique
outlined
t%
processing
total
multi-component
MMIC A u t o m a t i c
The
during
maintain­
to
shorted
defects.
increases
circuits.
'RF a n d
Ga As
due
moni-
-gases.
l'
related
Typical
economically ,
functionality
is'
chip.
masking
by
deposition
r
the
or
ofi n t e g r a t e d
improved'
the
source-to-gate
The p r e s e n t
controlled
FETs
that
defects
a deposition
on
failures
photoresist
using
is
of
GaA6
thedevi ce
GaAs
general,
such
yield
l e n g t h of
for
3 0-
accurately
The dc
mo d e s
-
be
consistent
gate
7*
-assessment
will
significantly
costs
to
final
incurred
device
in
assembly.
measurement
and
of
Buck [ 3 1 ] .
system
The
has
system
DC c h a r a c t e r i s t i c s ,
S - p a raqpe t e r s ,
noise
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 .
figure,'
gain compression,
%
RF
test
equipment.
within
the
reversal-
reck,
is
desktop
RF
etc.
ma n a g e s
and
large
Prior
matically
to
the
pf eobe
at
wafer
the
test
of
performed
station.
equipment,
wafer
data
an
example
is
probe
step
A
sets
the
movement s,
and
produced.
measurements,
using
substrate
switching
wafer
the
of
a typical
channel-select ion/RF-route
the
all
quantity
the
shows
m
drives
calibrated
. This
2.10
equipment
controls
routing,
the
strate.
Th e
implemented
computer
signal
Fig.
the
system
is
auto-
integrated
contains
calibration
sub1
c o - p l a n a r components
k n o wn
v
whi ch
have
the
are
f o r me A
the
m easurem ent
with
a
on
an
cells
m inutes
to
measure
an
$
\
r
entire
Having
of - MMICs
ents,
the
in
next
Over
wafer
general,
chapter.
the
the
MMIC
described
the*central
the
woul d
the
of
technique
results
takes
m easurem ents
this
’
to
and
a l l o­ w s
be
ma d e ,
on
each
approximately
locked
thesis,
of
the
first
approximately
design, -fabrication
phased
tested,
tips.
It
take
be
S-parameters
S-parameter
100
to
T+»is
probe
wafer.
MMIC
topic
devices
substrate.
at
shows
GaAs
perform
as
error-corrected
plane
2.11
on
width
alumina
of
reference
Fig.
50
s a me
c e l l .
be
To
9 'hours.
and
oscillator
will
k
testing
compon­
discussed
*
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
Fig.
— :
7
2.10.
, 0n-Wafer
R.F.
Measurement
System
Hardware
........
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
[31]
T
-
33-
.IHl
!
S,2a
IB )
f
(GHi >
((,HJ-)
?
,
3
t » < i
-30
Fig.
2.U.
M e a s u r e d FET S - P a r a n e t e r s o f
I . C .n on a GaAs Wa f e r [ 3 1 ] .
f
the
First
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 .
50
i
CHAPTER
3
*
-»
PHASE
LOCKED OSCI LLATORS
'
“
In
phase
this
locked
’‘
e
)
chapter
X
s o me
oscillator
possible
are
configurations
presented
together
of
with
a
an
4
identification
of
each
the
system
components.
individual
building
block
of
*
oscillator
implementations
are
locked
Also
of
the
least
frequency
understood
divider,
are
components,
isolated
for
The
requirements
the
different
phase
**
discussed
at
length.
th e , p h a s e
further
d/tector
and
investigation.
i
3.1
System
Configuration
There
oscillator
the
t wo
diametrically
stabilizers;
first,
improve
are
a
the
narrowband'
narrowband
performance
is
accompanied
by
is
phase-locked
PLL
of
a
is
In
clean
the
and
as
oscillator
second,
reference
varieties
wideband
employed
another
noise.
to
opposite
to
a
a
or
stabilize
(32].
In
filter
to
signal
noisy
of
that
oscillator
the
locked
oscillator.
.i
‘ The
is
if
obtained
low-level
tor
performance
best
one
t wo
for
phaselocked
Bandwidth
consistent
taken
from
of
-*the
with
the
separate
long-term
to
the
loop
either
type
oscillators
stability
first,
should
maintaining
locked
for
for
be
as
reliable
and
of
are
a
stabilizer
used;
second
*phort-term
narrow
lock,
the
oscillator.
s
u
~
l
_ i. ;________ __:----:----- ____________________ __r ..____________
oscilla­
possible,
>
f
very
stability.
as
and
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 .
output
An
microwave
effective
oscillator
source
s u c h ^as
tracks
out
so
that
a
the
the
multiplied
output
to
stabilizing
d.
U sli^
microwave
spect r3l
The
has
harmonic
any
the
limitations
of
the
to
lock
locked
of
*he
low-frequency
Her e
the
stabilization
of
l ow
and
Four
oscillator
of
[33],
stability
signal.
with
purity,
is
a stable
of
frequency,
one
signal
of
oscillator
technique
RF
stabilization
the
lo o p
oscillator
frequency-
source.
this
an
of
fluctuations
oscillator
oscillator
a
crystal
reference
unstable
c,
to
phase
Another
v.
met hod
are
these
each
lock
s hown
in
techniques
and
individual
divide
the
reference
techniques
Figures
one
frequency
good
to
the
possible
excellent
noise,
is
for
3.1a,
can
b,
produce
accuracy,
high
frequency" s ta b i lit y .
configuration
is
in
.
realizability
the
of
the
elements
and
also
the
*
elements.
a
the
*
perff' orraance
of
1
to
Th e
17 p h a s e
locked
differint
system
loop
u
be
will
components
which
comprise
v
d e s c r i b e d in the next S e c t i o n ,
a
T
' •■■■■ t________ :___r.=-T-.__ ^ ___ _____ _______
___
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
~
-
R
- o
REFERENCE
I H ARM ONIC
O S C I L L A T O R — “ 1G E N E R A T O R
36-
r
PHASE
DETECTOR
,
7
■
*%
O
V_y
1-2 O H ,
RF
^
Y
O SC IL LA TO R
<Q>
MTH
MULTIPLIER
HA R M O N IC
XP
RF
OUTPUT
PHASE
r e f e r -l nce
oscillator
r— — — —
DE TEC TOR
MUL TiP l IER
XN
FC
(b>
1-2 Cm
Rl
OSCIL I ■■■"CP
RF
OSCILLATOR
RF
OUTPUT-
m u l t i Rlie r
xp
r
RF
OUTPUT
f t )
REFERENCE
OS CI LLATOR
t
PHASE
DETECTOR
FC
RF
CTCILLATOP
olitput
d)
DI V I D E R
1/N
I
PHA
REFERENCE
detector
H g,
3.1.
Phase
Locked
Oscillator
E
OSCILLATOR
—
Configurations
[34],
»
;
...-------- .. . , ,______ ,y . . ,- •
_
<r
.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3.2
System
Components
v
Phase
locking
locked
the
oscillator
• frequency
fundamental
to
of
operates
oscillators
at
a
(PLOs)
‘o u tp u t
multiple
highly
stable
a
lower
frequency
derived
frequency
or
a
are
of
submultiple
crystal
a
of
ocillator.
than
the
by
phase
microwave
the
output
The
crystal
microwave
os c i 11 a -
T
tor.
The
primary
oscillator,
balun
a
builrfing
microwave
combination,
phase
detector.
mo r e
sophisticated
^ sizers
is
the
following
3.2.1
a
flocks
of
oscillator,
multiplier,-
One
extra
a
a
locked
These
a
pass
which
is
are
divider
filter
and
present
oscillators
components
reference
frequency
l ow
component
phased
mixer.
a PLO'are:
and
in
a
the
synthe-
discussed
in
the
sections.
Oscillators
f
Oscillators
feedback
with
or
fai"l
negative
positive
into
resistance
feedback
three-terriinal
one
devices
also
may
be
of
t wo
types.
has
g e n e r a l c a t Ko r 1 e a :
%
A t ^ h r e e - t e r m l n a l FET
negative
placed
in
resistance,
so
category.
To
either
\
realize
the
oscillator
feedback
cavity
is
an
or
an
required
loop.
s o me
■’ T h i s
form
have
some
form
between
mechanically
mechanic^
tuned
or
cavity
may
of
be
is
in
If
frequency
required
the
form
the
of
in
a
oscillator
tuning,
a
selec-
\
electrical
may
be
circuit.
tC
tion
resonator
resonator
inductor-capacitor
to
of
as
tuning
simple
can
as fan
be
made.
A
adjustable
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 .
-
screw
in
a
probably
cylinder.
consist
An
of
a
38-
electrically
tuned
circuit
voltage-variable-capacitance
I
woul d
in
a
LC
circuit.
Many
different
account
before
topology
will
design,
a
be
considerations
decision
used.
consideration
is
ma d e
Before
mu s t
mu s t
on
given
to
taken
into
whi ch
oscillator
a
oscillator
selecting
be
be
FET
all
the
following
po i nt s :
•
Efficiency
•
Amplitude
noise
of
DC t o
RF c o n v e r s i o n
modulation
and
frequency
modulation
J
spectra
*
Lo n g
term
Tuning
frequency
bandwidth
Temperature
Ou t pu t
st^jility
sensitivity
p o we r
Frequency
of
operation
fc
«•
Cos t
Size
and
weight
Reliability.
Having
of
its
noise
selected
locking
tor
phase-locked
be
discussed
common
oscillator
characteristics
injection
in. a
an
in
microwave
the
this
can
be
oscillator,
or
loop.
Both
chapter.
oscillators
will
topology,
accomplished
by
placing
of ' t h e s e
However,
be
improvement
by
the
either
oscilla­
techniques
will
first
most
the
described.
±
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 .
0
-
Fixed
Frequency
Every
FET
39-
Oscillators
oscillator
has
s o me
form
of
the
bi'as
between
tuning.
Fr om
<,V
the
very, nature
source
of
controls
the
fixed-frequency
generally
Th e
LC
loop
ed
in
a
output
in
power
may
is
FET’ s
the
the
be
very
in
By
element
in
varying
the
important
as
the
to
well
have
as
predicted
in
( LC)
should
predicted
loop
bias
be
the
only
observ­
efficiency
and
(35].
information
resonant
and
a
circuit.
FET' s
accurate
the
For
resonant
However,
altered
and
determining
frequency
substantially
gate
characteristics.
feedback
circuit.
difference
the
inductor- capacitor
oscillation
between
a
the
dominant
S-parameters
discrepancy
results
an
we 1 1 - d e s i g n e d
It
the
of
oscillation.
changes
region
oscillator,
is
of
FET,
depletion
consists
frequency
mi nor
the
actual
loop.
on
Any
S-parameters
oscillation
frequency.
\
\
t
Th e
are
being
that
the
very
very
a
of
to
iterations
correct
LC n e t w o r k s
in
fixed
consistent
simple
few
resulting
advantage
design.
will
os d i l a t i o n
at
microwave
poor
noise
from
frequency
batch
The
to
frequencies
batch,
primary
probably
frequency
oscillators
have
is
to
be
l ow
Q'b
ma d e
that
w e l l 1 as
disadvantage
obtained.
have
as
is
is
before
Generally,
[36],
performance.
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 .
thus
-
Voltage
40-
C o n t r o 1 1 e<U O s c 1 1 1 a t o r s
4
y.
Many^
(
oscillating
able.
or
I
dc
be
used
in
tor.
latter
former
over
a
is
are
change
the
follow
of
_
the
control
to
using
in
for
but
a
is
limited
voltage
range
available
on.
to
YI G
in
by
[38],
a
while
Voltage
which
describe
the
can
tune
[37],
capacitance
The
the
low-
devices.
tuning
the
the
oscilla­
devices
slow
an
oscillator,
current
YI G
are
using
low-noise
or
finite
with
The
varactors
by
vari-
frequency
lotking
a
voltage.
concentrate
t wo
frequency
for
range
important
SI ew
oscillator
range
loop
be
discrete
functions.
resulting
bandwidths
parameters
electronically
a)
locked
a
in
‘t o
the
discussion
Controlled
( VC O) .
The
ma n e e
mo r e
tuning
will
Oscillator
by
be
possibilities
controlled
controlled
for
ma ny
tuned
is
source
oscillator
perform
thereby
device
varactor's
up
may
require
t r a n B i t i o f r betw een
the
phased
multi-octave
Th e
*
to
Oscillators
Th e
of
systems
microwave
variation
opens
reference,
the
a smooth
signal
oscillator
noise
to
be
Variation
applied
basic
may
electronic
of
frequency
it
states.
today's
frequency
This
steps
can
of
the
Rate:
can
other,
tuned
The
be
in
oscillators
rate
changed
at
which
from
one
response
to
a
the
perfo I
are:
the
end
frequency
of
the
change
parameter.
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 .
of
tuning
in
-the
-
b)
oscillator
the
/
is
settle
T i me :
The
within
s o me
static
ma x i mu m
time
frequency
a f1t e r / a
frequency,
taken
band,
step
by
the
centered
on
tuning ^signal
applied.
drift
Post
Tuning
measured
s e t t l'in g
ing
to
predicted
c)
cy
Settling
41-
time,
but
over
Drift:
The
a time
not
largest
interval
including
the
value
ljcrge
of
frequen­
compared
frequency
to
the
drift
s e ttl­
frequency
devia-
t i me.
d)
tlon^of
Tuning
the
Linearity :
experimental
The
tuning
m a x i mu m
curve
f r om'
the
best
linear
/
fit.
.
'M
These
noise,
and
electronic
parameter?
together
spurious-free
tuning
oscillator
with
response,
low
define
A. M.
and
the
broadband
requirements.
O
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 .
F . M.
,
—
D ielectric
i"i_
■■■■
Resonator O sc illa to rs
■1
111 1 ■■■"
■'■■■
1
M icrostrip
value
of
medium
100 to
do
lock
noise
can
exhibit
circuit,
the
on
performance,
oscillators
excellent
the
other
high
with
proper
noise
add
Q to
in
Q
this
or
high
to
the
temperature
a
on
added
Q results
frequency
to
fabricated
low
hand,
The
constant
^limited
resonators
[40].
and
nearly
generally)
D ielectric
frequency
provide
are
[39],^and
stability.
m lcrostrlp
and
circuits
200
not
temperature
42"
the
circuit
ver^
low
1
*
compensation
versus
temperature
characteristics.
*
A
line,
is
dielectric
shown
in
resonator,
figure
3.2.
mounted
That r e s o n a t o r
i
puck
t
of
high
compensated
Q,
high
material
dielectric
(ZrSqTiOi,
near
or
a
m icrostrip
consists
of
a
^
constant,
and
temperature
Ba Ti i +Os r ) .
\
Figure
The
a
parallel
D ielectric/ Resonator
3.2
equivalent
circuit
r e s o n a n t , RLC
for
circuit
a
in
dielectric
series
with
resonator
the
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 .
is
micro-
,
\
strip
line,
the
with
resonator
with
reference
[41].
a matched
Z -
the
The
load
Z
[l
plane
i mp u t
the
impedance
term ination
is
center-line
of
the
+ 2j
of
resonator
[42]:
- --------- _ 1
Qr
* i J
r
+
i
at
3 2
o
wh e r e
P
Q
»
coupling
factor
■ resonator
r
AF -
deviation
•» c e n t e r
*
*
Q
from
centet
^es onator
resonator
frequency
frequency.
/
/
3.2.2
Low P a s s
In
low-pass
the
the
phase-locked
filte r
is
VCO. m o d u l a t i o n
pass
filte r
the
loop,
/
In
has
input.
the
The
a
phase
transfer
for
detector
function
influence
means
Implementation,
on
the
output
of
the
and
low-
properties
modifying
the
a
of
loop's
[43].
a phase-locked
requirements.
due
to
external
as
oscillator
between
provides
ing
narrow
set
considerable
and
-perform ance
oscillator
First,
noise,
possible.
due* t o - —s i g n a l
V i.
Filter
to
the
loop
Second,
modulation,
th e re are
$
minimize
output
as
to
bandwidth
minimize
well
as
f
t j po c o n f l i c t phase
should
the
be
jitte r
ma de
transient
minimize
output
.
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 .
as
error
Jitter
due
to
internal
tracking
oscillator
and
acquisition
*
s h o u l d ' be ma d e
of
noise,
as
performance
wide
car t
as
thereby
properties,
possible.
only
be
the
achieved
at
some
improved
loop
bandwidth
Improvement
degrading
the
other,
consequently
I
t h e two is a l w a y s n e c e s s a r y .
3.2.3
obtaining
in
the
one
type
expense
compromise
of
between
Mixer
Mixers
urations
cant
are
[44],
used
"The
advantages
[45].
Among
in
use
of
o\ >er
these
a
number
a
its
*v
different
of
MESFET
mixer
conventional
advantages
is
the
PLL
presents
“d i o d e
configsig n ifi­
counterpart
possibility
"ol^pbtair-
a,
ing
conversion
further
gains
.advantage
o scillato r,
RF
separated
the
low
at
inter-port
Four
in
Fig.
aspects
results
*RF s i g n a l
off,
an?* t h e
noise< figures.
the . fact
and
IF
signal,
three
ports
of
the
the
most
Each
into
local
widely
of
MESFET' s
The ^sTingle
an
from
low
that
*are
MES F ET,
the
used
these
MESFET
mixers
because
mixer,
gate,
the
the
oscillator
source
linearities*
sim plicity
oscillator
is
then
frequency
and
This
of
of
ttfe
is
mixing
single
is
is
is
MESFET
to
from
construction,
but
using
mlwer
its
introduces
biased
superimposed
achieved
shown
different
3.i»(i),
applied
extracted
are
/
Figure
gate
mixers
the
has
near
pinch
source.
The
^raln.
The
the
■-
at
local
intrinsically
exploits
nonlinearities.
MESFET
difference-frequency
local
the
A
coupling.
of
the
with
signal
3.4.
of
associated
*
on
the
the
the
primary
dc
voltage
MESFET' s
non-
advantage
of
disadvantage
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 .
45-
18
the
port
r
between
poor.isolation
RF
i
[ 46 ] .
The
MESFET
mixers
can
^
t^ie
single
and
LO
ports
generally
be
as
wel^
as
categorized
'
IF
into
«
MESFET ' m i x e r s ,
Figure’
MESFET
mixers,
Figure
3.4(11)
mixers,
Figure
3.4(111).
The
3.4(i),
and
the
dual-gate
the
dual-gate
balanced
MESFET
MESFET
mixers
have
0
one
major
advantage,
properties
of
a
it
has
balanced
tion,
8 p u r i ou s - r e s p o n s e
[46].
The
well
as
balanced
a.n
circuit
and
The
component
is
isolation.
MESFET
mixer
are,
rejection,
hybrid,
Increase
next
LO/ RF
this
the
section
used
and
configurations
input
may
good
to
good
LO
noise
require
tends
The
an
to
main
isola­
re'jectibn
IF
hybrid
as
complicate
the
size.
discusses
transfer
the
the
m ultiplier;
crystal
frequency
this
up
to
9
the
o scillator's
3.2.4
M ultiplier
The
in
frequency.
the
gain
MESFET
microwave
instead
of
m ultipier
field.
the
has
Its
us.ual
gained
widespread
advantages
conversion
include
loss
attention
conversion
associated
with
■*
t
Conventional
dynamic
diode
range
mixing
in
m ultipliers,
[47],
a
The
FET,
linearities,
are'
application
the
§chottky'
created
as
[48]i
by
both
m ultipliers,
figures
than
r—*
first
pinch-mlf
although
single
gate
^ GS ~ ^ G S
in
the
*
of
diodes.
figure,
nonlinearities
the
used
• The
barrier
the
same
namely
,
low-noise
these
The
FETs,
t e n ’d
Figure
that
high
produce
a n d • I p s —^ GS
non-
frequency
m ultiplier
^
effects
results
from
second
e ffe c t, \ Dual-gate
they
and
to.
3.3,
-1
nonline&rity
F ETs
have
are
larger
[49].
y *
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 .
is
popular
noise
—
Figure
3.3..
Dual-Gate
a-p.
MESFET M u l t i p l i e r ,
IF
RF
«)
LO
*
t
Li )
LO
*
Ut)
H
y
e
R
i
LO
\
RF
Figure
fcfefc-.**,
I F
RF
..... . . —;---------------- ;--- ------;—
D
3.4.
1
1
c
0
t
f
N
FET M i x e r
e
1
I F
C
K
Configuration
-----
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 .
3.2.5
Frequency
Another
>
locked
is
the
phase
locked
in
the
classic
the
divider
is
area
circuits
processing
[53,54,55].
to
digital
frequency
basis.
frequency
bandwidths
can
been
a
these
one
of
individual
circuits
freqAncy
division
of
a
given
in
function
exploited
in
microwave
literature
may
viewed
The
first
a manner
yielding
signal
be
on
sim ilar
instantaneous
a
c y c 1e - b y - c y c 1e
*
«.
'
tive
of
the
in
v
*
be
ability
categories.
counters,
of
and
applications
the
concepts
basic
tjiat
counting,
different
described
three
systems.
[52].
number
concepts
phase
Frequency
Additional
signals
in
microwave
compress
of
includes
of
where
have
to
are
[50,51].
wideband
Each
belonging
variety
used
divider..
warfare,
to
of
commonly
electronic
f r e q u e n c y ‘‘ d i v i d e r
category
a
applications
of
is
frequency
in
oscillation
There
as
which
the
necessary
Among
lie
component
oscillators
division
9
Divider
The
second
divider
concept
has
major
category
circuits,
been
known
consists
illustrated
*since
of t h e r e g e n e r a f
Fig.
3.5,
whose
in
1939,
[56],
The
necessary
a
&
components
feedback,
include
and
In
regenerative
at
the
present
am plification,
fixing,
subharmonic
filtering.
briefly
reviewing
divider,
s u b h a r r a o n i c„
in itially ,
■*
transients.
This
it
of
is ^assumed
the
either
signal
the
that
incident
due
is
basic
to
then
operation
a
signal
signal
noise
or
v
mixed
with
of
the
component
frequency
input
the
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 .
signal
incident
-
48-
\
slgnal
Co p r o d u c e
subsequently
mixer.
The
a
lower
filtered
This
oscillation
yields
Is
a
sideband
out,
leads
to
the
loop
when
frequency-divided
turn-on
determined
threshold
'by
the
at
quiescent
subharmonic,
am plified,
and
fed
the
buildup
of
a
gain
exceeds
unity
counterpart
for
to
s u b h a r m o n is>c
loop
gain.
back
which
to
the
subharmonic
[57].
the
the
LOW PASS
MI XE R
the
Input
This
signal.
oscillation
is
AMPLIFIER
FILTER
IN
OUT
FEEDBACK
NETWORK
Fig.
3.. 5 .
.General
Concept
of
Regenerative
Frequency
Division.
The
18
the
of
occurs
Input
♦
type
of
the
diodes
frequency
param eteric
division
the
third
by
signal
is
used
the
by
frequency
Harrison
type.
a
In
of
nonlinear
approach
is
simultaneously
on
commonly
technique,
an
exact
capacitive
bas^d
[58],
category
this
generation
divider
to
divider
the
whereby
provide
frequency
subhamonic
circuit.
general
a
used
pair
both
of
This
principles
of
varactor
(param etric)
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 .
-
am plification
described
MESFET
in
nixing.
this
thesis
"However, the
and
characteristics
frequency
division
In
the,
and
gate
source
Fig.
3.6,
term inals
by
inductors
resonant
the
in
to
the
of
in
Fig.
accomplish
inputs
the
v
mixing
The
the
process
gate
is
frequency
FETs.
initiates
the
loop
shown
novel
3.5
circuit
relies
tjh<
on
paranet rlc
[50].
capacitance
harmonic
49-
at
is
described
are
Fx n / 2
applied
nonlinear
generation
term inals
produced
Fj^
of
to
gate-tothe
aub-
previously.
The
chosen
between
so
the
that
FE- T' s
a
gate
,•5
to
source
drain
capacitances.
term inals
of-phase
at
used
to
in-pha^e
\
resulting
signals
outputs
at
Fjjj
t
at
and
the
FET'a
are
ou t -
is
then
Fj j j / 2 .
The
I
are
The
balun,
extrtect
described
N
the
Fin/2
in
the
next
signal
section,
from
the
divider.
;
+ F.I N
-nrrrn—I
Fig.
3.6.
Frequency
Divider,
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 .
frequency
-
3 . 2 . 6*
B■ ai iiu n
:
A
balanced
la
circuit
which
transm ission
balanced
line,
unbaj/anc^d
3.7,
the
unbalanced
can
magnitudes
conductor
opposite
the
line
the
transm ission
transm ission
✓
out-of-phase,
the
B.
a
coaxial
that
of
is
the
creating
a
line
typical
conductor.-
current
on
the
balanced
Inherently
balun,
by
at
of
on
outer
wavelength
choked
is
At
the
short
port
A,
equal
the
e n d vof
conductor
is
^circuited
center
conductor current
v
& <*
transm ission
line
D.
lines
B
and
D
system.
Transmission
in
center
A
3.7.
shown
the
B
Fig.
(two
symmetrical
Starting
flowing
line
a
hasc u r r e n t s
transm ission
a
of
cable
follows.
Sim ilarly
lineA
example
direction
quarter
line
is
.an
as
outer
A,
transm ission
is
operation
outlined
and
result
line)
transm ission
but ®
by
transform ation
from
a
•V
•*
a unbalary:ed tr a n s m is s io n lin e
to
a
twin-lead
The
be
transm ission
The
line
whereas
(59].
Fig.
choked
performs
ba l u n .
^called
p a ra 11e 1-conductor
of
50-
Ling
Balun,
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*Ve
180°
-
51-
4 -
G&JLP r a l l y . ,
GaAs
substrate
limits
the
circuit
diagram
Fig.
with
3.8,
of
[60).
so
phase
shift.
such
that
between
phase
t
The
to
/rrb^j^e^ o f
by
*
used
optimizing
balun
the
and
0
The
is
0
and
Li
are
is
a
+9 0*
a
designed
there
for
the
In
quarter
is
a
chosen
phase
1 i d^ D
• 180
B ar
.
elements over
four
shown
B and
there
be
baluns,
is
has p o i n t s
can
plane
L2
C2
a
out
a
of
large
^
frequency
interest.^
The
is
e l e m e n t s Ci e
on
ground
balun
line
The
circuit
This
monolithic
-betweenp o in ts
J)
fabricated
feasible
that
each, o t h e r . This
*
idth
of
transm ission
point
resulting
are
backplating.
possible
The
long,
circuits
selection
a
-90°
shift.
gold
possible
wavelength
P
monolithic
in
next
most
section
p+ijse
describes
locked
the
phase
detechor
loops.
=
b
B
t
A
Hhrl)—' D
X
J
Fig.
3.8.
Monolithic
Microwave
Balun.
s
tiki,
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 .
which
-
3.2.7
Phase
. — —
52-
Detector
— — —
Two
— —
broad
.
v
categories
of
phase
detectors
can
be
a
distinguished:
7
detector
output
the
f 61 ] .
-
A
voltage
between
the
voltage
is
a
is
the
phase
of
of
the
average
the
two
phase d e t e c to r s
the
sequential
detector
function
crossings
m ultiplier
which
and
sequential
that
zero
comparison,
m ultiplier
produces
time
input
an
interval
signals.
produce
product
phase
of
a
dc
the
In
output
two
input
s i g n a 18.
, *
.
*
Sequential
with
digital
circuits
(rectangular)
form
crossings.
poorer
c?
waveforms
Sequential
may
then
the
They
in t h i s t h e s i s .
i
F o r an i d e a l
m ultiplier,
with
trigger
due
to
detectors
then the
UHF
implemented
operate
phase d e te c to r s are
lower
V
usually
result
phase
capability
Sequential
frequencies
and
[62]-.
errors
.
are
(flip flops)
noise-handling
-detectors.to
input
defectors
/
consequently
edges,
extra
'
phase
on
wave-
missfng
usually
and
or
have
m ultiplier
currently
band
binary
phase
limitd'd
thus
are
not
inputs
are
considered
jiusoidal,
product
of
the
the
of
the
phase
useful
amplitude
diff^pdlnce
A popular
in
Fig.
dc
3.9,
1b
of
two
the
[46].
/This
able
frequency
It
circuit
has
which
both
proportional
them
mu 1 1 i w l T e X , p h a s e
mixer.
ma ny
output
between
double-balanced
over
in
inputs,
to
and
both
the
the
cosine
[$2].
detector
’can
a
wide
ranges,
and
circuit,is
>also
be
used
bandwidth,
has
good
is
shown
as
a
avail'-
performance
[46].
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n e r. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
-
53-
- LO
ULSL$LSULSJLt
Jt t-J
IF
RF
Fig.
3.9.
M ultiplier
Phase
■ O t-
rf*
LO
3.10.
i
.
(Diode-Ring)
T
IF
RF
-O i-
Fig.
Detector
M ultiplier
Phase
X
Detector
(Diode
Rectifier).
:--------------- — :---------------
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A phase
in
Fig.
3.10
vector
sum
are
then
Th e
f
useful
and
difference
is
critical
these
the
if
t wo
dc
g'ood
a
pass
filte"rs
use
in
of
a
ature
a
to
be
very
circuit.
lossy
The
draln
This
es
coupling
occupy
phase
as
the
voltages
RF
detector
a
the
two
signals,
so
that
resistances
two
are
balance
frequenthe
rings.
-'Is
is
composed
diode's,
F ETs
the
modulated.'
and
large
and two
of
low-
difficult
phase
which
however
%
the
realize
*
detector
its
of
utilizes
is
shown
in
to
same
the
detector
jmodulated
by
corresponding
This
to
phase
proportion
according
are
which' requires
the
quadrchip
■
F ET’s ,
operates
s 1n g l e - b a l a n c e d
of
It
circuit,
detector
between
differen-
configuration
s i n g 1e - b a 1anced
monolithic
phase
two
declined with
[46].
configuration
t—
novel
the
At.low
and* p a c k a g e d
Schottky
rectifiers.
voltages,
phas e - d e t e ct o r
area.
A
which
•
detector
would
signals,
small
has
shown
the
between
avoid/ed.
circuits
Is
produces
diode
is. th e
circuit
3.11).
is
the
difference
phase' d e t e c t o r
feasible
hybrid
is
c ommon
input
rectified
of t h i s
transformer
form
by
output
dc
c o u p l e r , t wo
phase
/
a monolithic
could
larger
microwave
(Fig.
Any
two
t^ ie
the
very
transformer
signals
integrated
hybrid
wa s
the
is
offset
single-balanced
3 ' dB
dc
Since
popularity
of
of
signal
A popular
the
to
voltages.
between
once
Here t h e h y b r i d
output
-
ce
advent
[^2^.
converted
rectified
cies,
d e t e c t o r which
r
novel
dc
phase
source-toFig.
principl­
[46].
their
3.12.
The
gate
respective
drain-to-source
detector
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 .
offers
55-
IF
RF
Fig.
3.11
Single-Balanced
Phase
Detector
LO
IF
05
Fig.
3.12.
Doable-FET
P h rase
Detector
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 .
-
56-
size
and
p
the
advantages
the
single-balanced
this
novel
Chapter
3.3
of
small
phase
resistive
detector
performance
diode
Is
the
mixer.
major
comparable
to
A description
of
topic
discussed
in
5.
Discussion
of
MMIC P h a s e
Locked
O scillator
C o m ponejt-t
!Ji l ls
*
*.
*
Depending
#»
-locked
on
—
oscillator,
the
all
'
-specific
™
or
s o me
configuration
of
the
of
components
a
phase
that
have
A
been
discussed
will
oscillator
will
oscillator
[63].
either
the
be
Each
are
based
at
of
of
own
required.
a
to
the
be
the
, than
situation
locking
will
general
the
thjg
to
locking
reference
oscillator
divided
reference
the
will
frequency,
or
reference,
or
two.
the
the
In
frequency
this
the
phase
advantages
on
lower
Under
oscillator
a combination
its"
be
multiplied
locking
has
be
locked
and
oscillator
disadvantages,
components
configurations
the
realizability.
lim itations
In
addition,
8
they
j
all
lock
require
with
the
vo1t a g e - c o n t r o l l e d
The
frequency
design
tuning
reference,
capability
and
hence
MMIC
VCOs
maintain
require
a
)
o s c i ’l l a t o r .
of
to
/
is not
unlike
that
of
the
i-
hybrid
integrated
great
depth
tions
on.HHIC
[64,65].
appeared
[68].
VCOs
which have been stu di- ed in
a
a d d i t i o n , r e c e n t l y , s ome p u b l i c a ­
In
VCO \a a r e
Similarly,
have
circuit
available
MMIC
166 , 6 7 ] .
m u l t i p l i e r and
Again,
m ultipliers
mixer
and
publications
mixers
have
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 prohibited w ith o u t p e r m is s io n .
been
-
57-
4?
studied
is
for
a
great
required
in
the
The
discussed
a
or
using
[32,33].
texts
due
using
the
passive
circuit
have
specifications
to
to
of
be
a
are
mo r e
large
size
examined
an
MMIC
of
the
approaches
components
off
' thoroughly
in
two
particular
little
However
required
chips
and
MMIC c o u n t e r p a r t s .
At ‘ p r e s e n t ,
integrated
alternatives
their
of
arise
either
of
[66,68]
requirements
components.
ivailable,
years
filter
number
desi^n^~p^oblems
of
development
loop
in
p ^ a isiv e
number
on t h e
the
in
are
MMIC w a f e r
wafer.
light
These
of
the
p h a s e - l o c k e d - o s c i 11 a t o r
requirement.
In
few
publications
•detector
phase
a
the
As
greater
a
these
MMIC
components,
frequency
result,
this
understanding
there
divider
thesis
t>f
the
are
and
very
phase
concentrates
frequency
on
divider
detector.
Chapter
presents
with
on
[53,66],
developing
and
contrast
4
a detailed
describes
analysis
the
of
MMIC
its
frequency
&
performance.
divider
and
*
Chapter
evaluates
its
5
examines
the
MMIC
pf\£se
detector
and
performance.
/
r
'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 .
-
58-
CHAPTER
U
FREQUENCY HALVER
a *
w
*
Some
the
use
of
unknown
a
categories
erative
and
but
required
input
generative
divider
has
Inherent
stability
this
chapter
loss
novel
can
and
provide
c o n v e r s i o n l o s s and
■
‘
Using
the
division
their
a
digital,
A few
in
three
regen­
divider
is
microwave
the
literature
unstable
monolithic
frequency
performance
type
divider
required
a
frequency
to
has
of
halver
the
its
configuration
shown
large
but
a signifi-
turn-on
power.
introduced
problems
of
in
large
power.
technique
by
are
demonstrated
this
solution
turn-on
occurs
has
limit
parametric
a
with
the
large
t
usefulness.
The r e -
reasonable
has
This
along
regenerative
frequency
a
performs
there
bands.
noise
divider
problems
is
field'.
frequency
appeared
limited
A
divider
and
dividers:
have
demonstrated
conversion
frequency
has
parametric
3.2.5,
digital
inherent
power
halver
require
it.
frequency
dividers
-their
on
The
lower
{5 A ] .
The
frequency
frequency
frequency
performance
*• a
to
frequency
communications
Section
parametric.
(53,66 J ,
cant
in
configurations
The
the
analysis
of
generally lim ited
fc
digital
frequency
The
in
novel
discussed
general
15A],
dividers.
theoretical
As
oscillator
device
introduces
detailed
y
locked
frequency
relatively
chapter
phased
of
a
generating
param etric
an
exact
divider,
subharmonlc
}
/
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
the
input
circuit.
signal
A realization
strated
by
pair
of
transmission'
tion
[52].
are
using
The
chosen
cy,
the
to
t wo
can
be
covering
8-16
the
required
subsequent
four,
stages
mediate
sixteen,
together
the
to
is
input
is
the
frequency
frequencies
threshold
5-10
overall
necessitate
Th e
typically
approximately
an
frequen­
parametric
input
loss
lines
balun.
possible
(e.g.
achieve
etc.,
the
the
12
using
frequency
utilizes
with
FETs
eliminating
desired.
provides
the
dBm,
any
division
of
use
a
divider
GaAs
FET b
novel
which
in
circuit
is
of
place
presented
of
inter­
approach.
in
varactor
The
big
is
the
Also
the
need
the
excellent
output.
Using
p o s s i b i 1it y * o f
this
diodes
advantage
of
amplificabion
"semi-unilateral"
isolation
this
between
circuit
)■
conversion
if
MM—
—
suited
for
I
fabrication
in
gain,
further
the
approach,
of
resonant
such
a
‘-•__
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
is
the
FET
loop
and
divider
N
a MMIC v e r s i o n .
.............................................
thereby
division
property
-
well
dB
signal
»
of
a
configura­
necessary
the
load
am plification.
Th e
thesis
division
used
co-planar
since
demon­
transmission
a
conversion
Furthermore,
bridge
the
While
been
capacltively
half
provide
capacitive
has
at
wideband
the
initiate
eight,
to
mixing.
GHz),
of
resonance
quite
to
in- a ^ b a l a n c e d
lengths
diodes
band.
to
a
nonlinear
technique
diodes
coupled
varactor
response
this
lines
is
and
balanced
varactor
produce
amplification
a
-of
electrical
output
nonlineir
across
with
is
4.1
Frequency
Halver
Model
*
r
The
here
parametric
la , shown
effect
tors
Vp(t),
RB ,
an
the
represented
an
gate-to-source
conductance,
a
applied
nonlinear
largely
consists
resistance
of
the
FET.
equivalent
model,
a
Fig.
phase
4.1(b),
to the
of
the
of
the
two
capacitances
subharmonic
of
and
is
At
an
phase
amplified
appears
at
the
small
approximate
gates
a c r o s '^
by
output
input
in
the
device
sm all-signal
at
resonant
the
the
resonant
be
of
a
drain
frequency
where
the
diodes,
Cg ,
loj^p,
and
which
the
maintains
FETs.
two
two
the
This
sub­
transconductance ,
of
levels^
a
t h e F ETs
series,
term inals
signal
pu mp
two
can
\
consists^
inductors
the
FET
barrier
The
input
A
with
[67,69].
of
Schottky
subharmonicq .
gate-to-source
harmonic
the
Rs
associated
capacitor,
current
resis­
resistance
w h ic h
feedback
field
lumped
Each
loss
desired
out
Rl .
two
Instantaneous
two
the
capacitance
the
L,
considered
of
with
plus
to
in
produces
source
draln-to-source
Referring
is
loss
circuit
consists
Inductors,
capacitor,
and
and
output load
inductor
by
divider
pumping
Input
g a t e - t o - s ou r c e
2to 0
4.1(a)
a
two
and
represents
the
InF ig.
transistors,
voltage
frequency
g m,
FETs.
resonant
frequency
loop
u> 0 ,
has
which
j
in
the
b l^ s
case
voltage
. u) 0 As
of
Fig.
4.1(b),
an
1//L
will
output
with
zero
g a te -to -s ource
is
be
Cg ( 0 )
(4.1)
shown,
4
of
and
frequency
u
param etric
*
subharmonic
generation
/•
(and
its
o d d —o r d e r
harmonics)
t
.
......
;
... . „/
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
.
can
-
61-
9 l
A / V t ^ - t j t y w v V '.
Jrr. V;I
(V
Figure A.I.
Field
(a) P h y s ic a l c i r c u i t
E ffect T r a a s ls to r Frequency Halver:
la y o u t; (b) E qu iv alen t c i r c u i t la y o u 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 .
•-
62-
*
be
obtained
band
in
extending
2(i>0 .
The
the
input
The
shown
Fig,
Formulation
The
nonlinear
[69].
of
charge
q(v)
is
a
above
depends
power
formulate
the
behaviour
the
D ifferential
on
required
differential
of
the
frequency
for
Schottky
defined
Equation
capacitance
generating
gate-to-source
-
slightly
divi-aion
input
in
2u0 .
will
the
to
2u> l y i n g
A.1(b).
component
me t a 1 / n - t y p e
below
minimum
gate-to-source
The
layer
*•
section
frequency
frequency
equals
describe
in
input
of
The
frequency
that
an
considerably
level.
next
equations
to
bandwidth
power
the
halver
from
exact
input
when
4.2
response
the
the
dominant
subharraonic
capacitance
barrier
as
is
varactor
of
a
with
signal
FET
a
is
a
depletion
[70]:
- A / 2 e £ ND (4>Q+ v )
(4.2)
where
i
c
A is
the
Junction
e
is
the
charge
e
is
the
perm ittivity
$o
v
is
is
Np
the
the
is
the
on
thermal
total
an
electron
of
semiconductor
equilibrium
reverse
donor
4
area
built-in' potential
voltage
concentration
m aterial
across
on
the
junction
n-side.
,
*
It
is
conventional
defined
in
«
convenient
to
quantities.
terms
of
the
express
The
varactor
the
charge
Junction
q
in
capacitance
charge-voltage
of
is
relationship
r
j
terms
a 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 .
-
63&
7
A
Cg( v ) - - d q / d v
Hence,
from
(4f3)
r
(4.2):
C ( v ) - -I A / 2ee N
8
^
DThis
relation
mathematical
model
for
F i g u r e £*4.2
network
the
For
the
shows
the
nodal
the
loop
vp i2
1
/ r « o
•
<*•*>.
starting-point
frequency
for
deriving
the
halver.
the
frequency
halver'a
the
n|cessary
v o 11 a g j
equivalent
j i r ops
and
voltage
drop
currents.
right-hand
side
of
the
network,
gives:
L * d / d t ( i 2 ) + Rs
Cp * d / d t ( v o 2
■ C g ( v 2 )• d / d t ( V 2 )
r'
I
‘
i 2 + *2
■ Cg( V2 )• d / d t ( v2 ) -
i22 i2
the
representation,
identifies
around
is
-
(4.5)
i 22
(4.6)
_ v2 )
(4.7)
“ C f d / d t ( v 0 2 -V2 )
’
(4.8)
wh e r e
Rs
is
the
resonant
loop
Cp i s
the
feedback
capacitance
L is
resonant
loop
resistance
inductance
is
the
],o a d
g Q is
the
FET dr *a i n
gm is
t h e »FET t r a n s c o n d u c t a n c e .
Combining
vp -
the
series
(4.5)
-
conductance
(4.8)
to
source
v 2
conductance
yields:
L • d / d t { Cg ( v 2 )* d / d t v 2 ) + &B* i C g (
►
Cp»d/dt
( v o 2 “ V2 >)
)* d / d t ( v 2 ) - C p « d / d t ( v o 2 - V2 )} ( 4 . 9 )
■f V2
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 .
-
For
around
vp -
the
the
l i f t —h a n d
loop
L* d / d t
64-
aide
of
the
network,
voltage
glvfa:
{ Cg ( v i ) • d / d t ( v i ) - C F * d / d t ( v 0 i - v j )}
♦ Rs* i ^ g ^ v l ^* d / d t ( v j )
-
(4.10)
CF * d / d t (v o i - v i )}
+ vi
*
»
\
0
•
«^V
Focussing
nuRt
sum t o
gL*
on
0 -
the
vq2
node,
“ V( ) 2 )
( v 0 1
"
sum t o
Cp* d / d t ( v 0 2 - V 2 )
at
the
zero:
voi
signal
of
Is
(4.9),
this
node
and
the
<
(4.11)
c u r r e n t at
t h i s node
the
t
by
( 4 . 12)
+ c • d / d t ( VQ1 - VJ )
(4.10),
by
(4.11)
and
halver.
fr e q u e n c y
facilitated
exists,
~ g mV2 ~gOV 0 2
node,
4j'O p e r a tio n
equations
c u r re n t at
,
g L* ( V y 1 - V u 2 ) +g 0 v 0 1 + g ffiv l
Equations
the
the
zero;
Similarly
must
drop
assuming
manipulation
and
(4.12)
Solution
that
a
of
define
these
subharmonic
n o r m a 1 l z a t 1 on
of
the
va r l a b l e s .
3
**■
-----------------------------------------------W
__1l
\ A ---------------------------------------------
L
Ri
-nm _A A A -
r - r
.
9P2 .
l.
a*
’a
3,
f
421
i
i
vp
I
9 » vi
Figure
4.2.
F ield E ffect T r a n s i s t o r Frequency Halver
V o l t a g e and C u r r e n t I d e n t 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 .
Nodal
-
Assumlng
thXxgate
to
frequency
signal
ing
higher
frequency
source
order
division
nonlinear
plus
65-
a signal
harmonics
occurs,
the
capacitance
has
at
half
this
vollPage
a'
across
fundamental
frequency.
Ignor­
f55 J :
S
+
vs
VR +
vo
VQ
V
+
+
VGS
<z>
-
<4.'13)
<
o
V2
'
° M .
1
V
01
>
V
O
VR
"
v r
are
s i-g-D- A
•
vQ
are
s
VGS ’ VDS
From
.to-source
+
’
a re
the
ign a
d
c
VDS
s
at
1s
at
ri^i
hi a s
de £i n i t
f .a
voltages
will
Thus,
equations
frequency.
%
)
vo
VQ »
vs
]T
> \*+ v Ga
be
in
resonant
loop,
anti-phase
(4.9)
and
(4.10)
the
at
the
two
gat e -
dividfed
become
L *d / ^ t { Cg ( v Q + Vs +VG S ) ' d / d t ( V Q + V8 ) : X:F * d / d t CvR - VQ + Vo - \ )
Rs * ^ Cg ( V Q+ Vs +VG S ) - d / d t (
y ^ fv 8
)-Cp.d /d t
)
(4.14)
( v R- v Q + v o - v g ) }
**
,
*
+ v^+v
Q
s
and ,
vp
+
.
L - d / H t l C g ' ( v q - V s +VG S ) . d / < i t ( V q - V B)
R. ' f Cg'( V
v 8 +VG S ) •< l / d t ( V
V
'
V
-
Cp - d / d t ( v R- v Q- V o + v> ) )
d / d t ( V *
'
V
+ v —v
Q a
■
■.
.
.
W
1
:
’
(4.15)
-
•
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
■
-
and
equations
( A . 11)
and
66-
( A . 12)
become
0 “ * L - ( “ 2 v y - C F - d / d t < VR - vQ + Vo ' Vs ) _ g n , , ( VQ + Vs ) - 8 o ‘ ( V R + Vo ) ^ V 1 6 )'
«
0
- *L- < - 2 V * V d/tft<virvQ-vo-H;vt.>+8m.( v V t g 0’<VR-V0> (4.17)
The
source
nonlinear
capacitance,
element
whl c - h
Is
in
a
these
equations
function
of
Is
the
the
gate-to-
fundamental
and
4
subharmonic
Given
Include
all
signals,
as
well
a
pumping
the
conditions
*
*as
voltage
of
vp ,
necessary
the
bias
equations
for
condition.
( A. 1 A) —(Jt). 1 7 )
realizing
a
sdbharmonic
signal.
J
To
a8
being
the
dependent
Independent
fermi
simplify
tha^
of
on
the
are
mathematics
the
pum^^Ujjg'
pumping
alike
to
be
we
will
voltage
voltage.
combined
Assume one
t
an) l a n o t h e t
This
for
procedure
further
as
being
will
allow
sim plification.
A ddlng,(A .lA )and(A .15)gives:
/
equation
*
2 v p - L . d / d t l Cg ( v q + vs . V G S ) . d / d t ( v Q. V 8 ) . C g ( v q - v s . V G S ) ^ d t ( v Q- v s ) )
-
-
L * d / d t { C * d / d t ( \) 3- v
e
K
if
+v -v
O
8
)+C
r
«d/dt ( u -v
. K
if
R , - l C F - d / d t ( v R - v Q* V o - UB) . C F - d / d t ( v R - v Q - V o + ^ )
*
2uq
Subtracting
-v
O
+,v ) }
}
(A.1A)
from
( A . 15)
yields:
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 .
(4.18)
-
67-
0 - L - d / d t f C g ( v Q + V8 + VG S ) . d / d t ( v Q + vs ) - C g ( v Q- v . V v G S ) . d / d t ( v Q- v s ) }
-
L - d / d t {C f ( v r - vq + vo - v s )
- C F - d / d t ( v R- v Q- v o + v B )'}
+ R3 * ^ Cg ( V Q + V51+VG S ) ’ d / d t ( V
/
.
-
'
>
Rg . { C F - d / d t ( v R- v
V
' V
V
V
VGS> ‘ d / d t ( VQ - Va > f
+ Vo - % ) - C F . d / d t < v R- y - V o + Vs ) }
'
+
(4.19)
4..
8
4
Adding
( 4". 1 6 )
to
(4.17)
48l ' , o + 2 g . > + 2 8 o ' ,o
produces:
" l CF - d / d t < V
V
V
V
- ^
’^ ' ^ R
- V
V
V
I
(4.20)
-Subtracting
(4.16)
from
(4.17)
delivers:
>? s „ v 2« ; u R " i v d / d t < v R - v uo - vs ) ' V
In
equations
(4.18)-(4.21),
gate-to-source
capacitance
including
feedback
tions
the
can
varactor
be
greatly
capacitance
In
order
to
have
d / , i ': ( v
the
been
given
the
Wh i c h
this
form,
the
gate
(4.4)
is
i( 4 - 2 , ) '
include
' I n,
once
dc
v
as
by, e q u a t i o n
separate
•
terms
v
combined;
capacitance.
sim plified
y
well
qb
the
to
the
terms
equa­
source
included.
gate-to-source
capacitance
«
from
the
small
capacitance
we
signal
nonlinear
*>
to
£he
overall
let :
VTg - V vcs
rk"
contribution
-
<
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4*22)
-
68-
He n c e
Cg ( VGS + Vo > “ 1 A / 2 e E N D *
where
v Q Li
a
small
signal
[ VTg + V0 r *
voltage
^
9
Defining:
V g ' WGS' - oo ’
c
we
(y r J
g
GS
■
47 *
J/
- yA /
2
/*
VTg
/
•
Tg. r *
tL. ^ vo v T
2ecNn
-r— ^
VTg
•
capacitance,
second
where
terft i
To
is
the
simplufy*
the
the
j f i r s t "term
small
signal
(4.25)
total
isth e
dc
nonlinear
the c a l c u l a t i o n s
Using
• g
m
terms
as
this
approximation,
' n:
ah o w
•* e
*
-
-C
of the
^
Q»
can
and
( v - v '+V
g . Q s G S
the
four
differential
necessary.
/
V•
T8
modify
the
* *
( 4 . 26)
follow\
) • d/d t ( v - v )
Q. s
**
v
(V p JW r1 g
GS
IV
9
g
capacitance
‘
C ( v + v +V
) • d/d t ( v +v
g
Q s . G S
Q s
*
we
gate-tfl-source
contribution.
e q u a t i o n s the ap p ro x im a tio n given below is
*
1
f l + v / V T I * ' . - [ 1- v / 2V
1
if v
%°
8
°
t 8
°
c
-(4.24)
" Cn
c ) , l l + v on / V rTgJ ' i
g ( V GS
M u a t i o n ■' s ( 4 . 2 5 ) ' d e f i n e s
'
<4 - 23)
have:
c g ( V rGS
c 4 v oJ
^
f
v r>
* d / d t ( v . )- 2 - d / d t ( v )+
Q
8
Tg
*d/dt(v
'
8
)}
(4.27)
•
■
v +V e ) • d / d t ( v_ + v ) + C ( v „ - v +V ' ) • d / d t ( v - v )
s
GS
Q s g Q s G S
9
s
}
V
■ C (V
) * {2 • d / d t (
8
GS
Q
V
* • d/dt(v_.)— » d / d t ( v )}
. VTg
,
Q - v Tg
,* .
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.
(4.28)
-
69-
Cp * d / d t ( v r - vq + vq - V s ) - C p * d / d t ( v R- v Q- v o + V8 ) - 2 C F « d / d t { v o ~ v s }< 4 . 2 9 )
V ,
CF - d / d t ( v R- v Q+ v o - v s ) + C F - d / d J : ( v R- v Q- \ f o + V8 ) - 2 C F * d / d t 1 v R- v Q } (.4 . 3 0)
These
defining
terms,
terms
equations
and
factors,
are
common
the
( 4 .1 8 ) - ( 4 . 2 1) .
normalizingthe
we c a n
to
obtain
substituting
equations
g'featly
frequency
with
simplified
the
halver
in
these
following
frequency
halving
equat ions .
Wenow
introduce
the
following norm alizations
and
definitions:
w p
1//LC
g
( V
)
G S
u> R
8 * .u>. / a) ^
p
R
Y “
X “ V„p / V £Tg
Y •" - V„o / VTo
Tg
1//LC~
F
C _ id
F p
Q p
v-uj/oo
l/(«
p
C (V
).R ) '
g G S
s
p
(4.31)
->
S * V«/*VT«
s
Tg
R "
VB/
R VT„
Tg
r
Q ’
VQ/ V Tg
T " V
In
addition,
represented
(4.30)
Into
ai
"
2 g L+ g o
differentiation
usjing
Newton's
equations
dot
°2 * gm
with
respect
notation.
(4.18)’ -
(4.21)
9
resuiting
2X «
four
nonlinear
frequency
halving
03 "
to'
go
will
t
Substituting
(4,27)
and
normalizing,
i
equations are:
o2
. 2
2Q- Q - Q Q - S - SS%-( 2L a 2 w ) Q+( 2L a 3u>p ) R
+ (2/Q
)Q-<1/Q
)QQ-(1/Q
) S S + ( 2 R ^ a 2 )Q
(4.32)
+ ( 2 R g a 3 ) R+2Q
/
4- .
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 .
be
-
the
0 -
2 S - 2 S Q - S Q -Q S , + C 2Lai i Dp ) Y+( 2a2(Dp L + 2 / Q p ) S
%
.>
-
(I/O
) S Q - ( 1/ Q
P
P
¥
0 -
) S Q + ( 2 a i R o ) Y + 2 ( l + a 2RD ) s
8
.
8
%
a i Y+ a 2 S+YY- YS
f
'
o
“0 - a 3R+a2Q+ YR - YQ
,
(4.33)
*
/
^(4.34)
* •
*
’
(4.35)
where:
X 1 b’ t h e
normalized
f r e q un c y
Q is
the
2v .
the
forcing
forcing
varactor,
of
function
angular
the
normalized
output
S is
the
normalized
subharmonic
angular
Y is
the
frequency
normalized
frequency
b e uslgd
a, f u n c t i o n
source
the
pumping
frequency-2v.
across
subharmonic
signal
barrier
which
the
varactor,
voltage,
of
evaluate
characteristics
diode.
'
forcing
function
j
the
pumped
subharmonic
across*
equation
(4.33) is
/
nonlinear
its subharmonic.
the
the
will
the
as
gate
be
tran sfer
to
called
resonant
Equations
and
of
(4,34)
loop
the
and
*
field
-effect
the.
will
output
■>
~ This
the
m
for
Ue
^ ,th e
m a g n i t u d e of
Equation
to
•
angular
'* ■
being
describes
voltage
equation,
output
equation.
jf r e q u e n c l e s .
*
fundamental
2v.
angular
voltage
(4.32) contains
» •
the-
*
(4.35)
of
developed
v.
to e v a l u a te the
Schottky
equation,
angular
v
of
driving
voltage
frequency
voltage,
v.
Equation
will
of
*
R is
of
function
'
normalized
across
input
call
transistor's
fundamental
equations
equation
and
and
(4.34)
the
load
driving
siibharmonic
a'nd
(4.35) the
*
subharmonic
output
respectively.
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 .
-
71-
4. 3 A n a l y t i c a l . S o l u t i o n
To
analytically
ential
equations,
def ined
as :
X -
X( o ) s i n ( 2 v
t
developed
and
are
equated
frequency-halving
let
the
mental
output
6u t p u t
voltage
forcing
differ­
function
Q,
by
begin
at
[71]
substituted
the
a
be
where
the
solved
the
the
real
approximate
differential
and
imaginary
by
first
equation
function
of
fundamental
is
the
of ^ t h a
frequency-
solving
the
used
obtain
to
voltage
fundamental
frequency
respectively,
them.
For
complex,
and
ha^
simplicity,
rather
t
I
assume
than
funda­
the
across
the
frequency.
*
(4.35)
gives:
ct2Q + at3R » - Y [ R- Q ]
will
using
zero.
derivation
This
equation'
then
into
of
to
the
equation.
, Rewriting
.We
be
(4,36)
can
Hayashi
equations,
as
a) / u>
P
equations
separately
R
v -
coefficients
differential
varactor,
the
where
are
the
We ' w i l l
halving
fitst
0)
- 2
solutions
equations
j r er ms
we
differential
approach
periodic
solve
’
Th e
an
&
(4.37)
that
R and
with,
a
will
their
Q are
amplitudes
phase
we
t
sinusoidal
R( o)
difference
re-present
sinusoidal
of
R and
at
and
$
the
Q(o),
between
Q in
^their
form.
’
Using
the
fact
that:
(hr
+jx
sin(x)
-jx
“ 2 j
Let:
. .
where
R “
■
.
- 2 j R( o ) ( a 2- a* 2 )
1 j ( V T + (j)) ■
a ■ — e
.
*
•
(4.38)
.
v
•
i
* d e n o t e s t h e complex c o n j u g a t e
__________________ !____ a___ _________
■, »*’
”
■
'■ 1'
'
■ ■• •
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1
». #
'ff
-------------
Similarly:
*■
Q -
- 2 J Q ( o ) ( a 2 e " J 2<t’- a * 2e J 25>),
Substituting
f
0 -
Into
(4.37)
- 2 J a 3R ( 0 ) ( a 2 -a* 2)
- 2 j a 2Q ( o ) ( a 2e " J 2
a * 2e j 2 *)
- 2 j Y & ( o ) ( a 2- a * * 2 )
,+ 2 Y R ( o ) ( a 2 + o * 2 ) ( 2 v + J )
(4.39)
+ 2 j Y Q ( o ) ( a 2 e “ J ^ - a * 2^
2 *)
- 4 v y Q ( o ) ( a 2e ‘ j 2 * + a * 2e j 2 *)
In
8teady
this
state
solution.
ft
e
R(o)"0, Q(o)"0,
fifth
terms
in
No w,
terms,
analysis
and
The
o
$ *
equation
since
a
we
2
0,
only
concerned
steady <s t a t e
thereby
solution
eliminating
with
requires
the
third
and
coefficients
2
a*
must
of
2
o
are
equate
independent
0 -
ja
3
to
zero.
give:
c o e f f i c i e n t s of
2
a*
equation
(4.40
)
(4.40)
give:
R ( o ) + J a 2 Q ( o ) e J 2 * + 2 Yv R( o ) - 2 Y vQ( o ) e J 2 *
Multiplying
and
non-zexo
0 - „ - j <*3 R ( o ) - j a 2 Q ( o ) e ' :1 2 * + 2 YvR ( o ) - 2 yvQ ( o ) e ~ J 2 *
The
the
4.39.
th e ir .coefficients
The
are
by
e ^ 2 ^ and
solvibg
'
*(4.41)^
yields:
*
W3o2-
j2
.
5 ( 0)
,
1 2yv-j a 3
Rearranging
equation
(4.41)
2
Y -f-j
2) Q ( o ) e
‘( 4 42)
(4.42)
provides:
J £♦
(
.
i
>
-
( 2
Yv+ j w 3 ) r ( 0 )
(4.43)
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 .
-
Using
(4.42)
and
(4,4
we
73f
have :
2
1 + [ q2/ 2 y v ]
-
------------------------ r
1+Io3/2yv]
Q (o)
arid
2$
« t an"
1
[ a 2/ 2 y v ] +t a n -
Equations
fundamental
R( o>)
as
output
on
output
[ a 3 /2 y v 1
1
and
of
for
the. normalized
capacitance,
y;
and
are
the
Q(o).
conductance,
solved
function
signal
obtain
of
b
£he
varactor.
steady
Here-
S(o)
is
the
value
solution
of
s hows 0 3
;
a
the
FET
feedback
fundamental
angular
by
n.
is
the
of
Y(o)
subharmonic
difference
equation
can , s i m i l a r l y
value
subharmonie
phase
Solving
equation
state
Y(o)
the
TJie ' a s s u m e d
represented
state
normalized
normalized
output
the
S(o).
and
steady
of
ss
subharmonic
to
solution
transconductance,
frequency,v.
, The
the
T^his
FET
a2;
the
(4.45)
(4.45)
equation
a , function
^dependence
FET
<4.44)
<4.«>
as
a
output
signal
across
between
Y and
(4.34)
be
the
S
is
provides:
2
y2( )
with
Y -
Y(
V i.
.
.
.
.
.
-
S (o)
l + [a2/nv]
n * tan-
1
0
t a 2 /yv]
X a - a * ), a n d
S(^b)(oe
i
l + [ « 2/ yvJ
.0
-
. _ .
*
+ ta n '
$e^
1
(4.46)
lai/yv]
VT'f n *
^ T1f - a * e ^ T1)
.
0
. '
•
R e p r o d u c e d with p e r m i s s io n of t h e co p y rig h t o w n er. F u r th e r r e p r o d u c tio n prohibited w ith o u t p e r m is s io n .
-
Thus
far,
subharmonic
Che
output
solutions
74-
fundamental
equation
s hown
in
outpuf
have
been
equations
equation
solved
(4.44)
and
and
and
the
their
(4.46),
respectively.
In
resonant
and
the
loop
equation,
Y( t ) eJ h
imaginary
derivations
in
-
j2$
^
we
complex
terms.
R( o )
of
will
/ l
.
♦
driving
require
number
Rewriting
Q ( oo ))
the
equation
the
notation,
equations
and
the
terms
RtoJeJ^^
with
real
(4.44),
(4.42)
M2 '
// T 1- Tv - Sv r -
and
gives:
(4.47)
and
j t a n _1M
jtan-1H
'
(4.48)
whe r e
M “ a
H *
Using
the
2
/ (2f v)
0 3
/ ( 2yv )
identity:
e j t a n ” 1X
we
hive:
/ ( M 2 + l ) ( H2 + 1 )
V( M2 + 1 ) ( H2 + l )
M
/ Tm2
?n
‘
expanded
r
proceed
^
to
+1
(4.49)'
)(H2
equation
into
its
express
+1
)
(4.49),
real
the
and
*
it
/
( M2 +1 ) ( H 2 +1 )
is
observed
imaginary
quantity
Jt(o)e2 j^
terms,
as
that
*
We
shown:
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 .
ej2^
can
is
now
-
R ( o ) e ^ 2 ^ * Q( o}
Similarly
for
Y(o)ej " -
75-
+ j QCo )
Y( o)
1
+0
as
a
}
(4.50)
l+O2
2
function
S<4>)
.
of
jS<o>
S(o)
{ i^ -|
1 +P *
. 11 J
1+P2
where
P * a i / yv
L “ a 2 / yv
(4.52)
Using
the
above
expressions
for
R(o)eJ,2$
and
resonant
loop
/
Y(o)eJn,
we
equation
(4.33).
behaviour
will
of
now
This
the
proceed
to
solve
equation
will
frequency-halver
V
the
determine
in
the
the
nonlinear
resonant
'
loop.
*
9
This
loop
consists
of
t wo
varactors
in
series
with
t wo
inductors,
V
Rewriting
equation
(4.33)
0 *= 2*S- 2SQ- SQ- Q*S- ( 1 / Q ) S Q - ( 1 / Q ) § Q + 2 ( a 2 w L + l / Q )& + 2 ( a i « o L ) Y
.
P
P
P
,P *
P
?
♦ ^(c^R
i
.
)Y + 2( 1 +<x2 R ) S
'
(4.53)
s
Assumi ng
„
subharmonic
behaviour
-
and
•
using:
'
(4 .5 4 )
%
*
we
c an
wr i t e :
>.
,
■•
S -
-
Y -
’ '
- Y ( o ) ( a e ^ n- a * e ’ :’ n )
Q
- 2 j Q ( o ) ( « 2. ‘ j 2 * -« * 2 . j2 *)
-
S( o )(a- a*)
' 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 .
7
where,
S
Is
Che
normalized
varactor
Y
Is
the
Is
the
angular
normalized
angular
Q
of
subharmonic
frequency
frequency
output
voltage
across
the
v,
subharmonic
voltage
of
v.
normalized
foreing
function
voltage
developed
t
across
the
varactor
of
Substituting
in
JT
solutions,
and
solving
for
demonstrated
* 0
-
2j v
previously,
we
angular
these
frequency
2
v.
approximate
the
steady
state
periodic
solution,
as
A
have:
S( o ) ( a-a* )
-
8
v 2 S ( o ) Q ( o ) ( a + a * ) ( a 2 e ' 2 * + a * 2e i 2 * )
t *-
8
v
2
P ( o ) Q ( o ) ( a - a * ) ( a 2e
^
a* 2e J
2
2
-
2 v 2 S ( o ) Q ( o ) ( a - « * ) ( a 2 e " :)2 ^ a *
+
4 j v / Q p » , S ( o ) Q ( o ) ( a - a * ) ( a ^ * ” ^ 2 ^,+ a *
+
2
2
^)
e J 2 ^)
2
e^
(4.55)
2
J v / Q p • S ( o ) Q ( o ) ( a + a * ) ( a 2 e ■ :1 2 't' - a * 2 e ; , 2 ',' )
c
+
2
v S ( o ) ( a 2 «>p L + l / Q p ) ( a + a * )
t
+
2
v( a i i i ) p L ) Y ( o ) ( a e ^ n + a * e ^ )
-
2
j ( a 1R > Y ( o ) ( a e J n - a * e ~ J 51
»
8
- 2j ( I+ a 2 R ) S ( o ) ( a- a* )
i
<
8
*
*
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 .
«
Since
the
higher
coefficients,
0 -
only
coefficients
order
har moni cs can
of- t h e
a terms
of
a and
a* a r e
be i g n o r e d .
of
interctt,
Solving
for
the
gives:
[ i v 2 ^ j i v / Q p ] s ( o ) Q ( o ) e " J2l p
+ 2 v ( a 2 u> L + l / Q ) S ( o ) + 2 j v 2 S ( o ) - 2 j ( l + a 2 R ) S ( o )
P
P
8
(4.56)
+ 2 v ( a 1 a> L ) Y (*o ) e ^ P - 2 j ( a i R ) Y ( o ) e J n
P
Using
'
equation
(4.51)
Y ( o ) e j T1 -
(
S
we
(
V
have:*
S(o)
+ j
(4.57)
1+P2
Upon ... s u b s t i t u t i n g
1+P2
this
value
for
into
equation
«
( 4. ^5
6
) we
-Q(o)e-'
obtain:
2
^ [ } v 2 - $ j :v/Q
P
' ] « 2 v ( » a 2 w L + l / Q ) + 2 v ( a i m L) [ ^
P
P
P
i+pz
+2 a j R [^ ~ —z ] ]
8
1+ P 2
]
(4.58)
+ j f 2 v 2 —2 ( 1 + a 2 R ) + 2 v ( a j u L ) [■—— — ] ]
8
P
1+P2
“ J [2oiR
1+P2
8
A similar
ients
results
of
exptessioju results
the
in
a*
the
terms.
solving
Combining
following
2
Q2( o) [ ( * v2>
upon
these
for 'the
t wo
coeffic­
expressions
equation:
2
+ ( W Q p>
] * Kq2 + K !
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 re p r o d u c tio n prohib ited w ith o u t p e r m is s io n .
(4.59)
f
-
78-
v
where
Ko -
[-j—
2 v( a 2 Wp L + l / Q p ) + 2 v ( a i u p L)
L+P
j + 2 a i Rg
]
1+ P
*+P
( 4 . 6 (I)
Ki ■
v Z-
2
( a 2R +l)+2v(aito
2
L)
] - 2 a^R
[
1 - PL
1+P;
1+P
and
-
^ ■ ta,n-
2
1
[Ki/Ko J + t a n -
Equations
solution
to
(4.59),
the
(4.59)
the
parameters*
voltage,
state
of
Q(o).
is
and
and
show
the
the
provide
In
by
at
the
between
forcing
demonstrate
an
that
function
the
angular
FET
the
particular
relationship
varactor's
varactor,
limited
(4.61)
equation.
equations
across* the
amplitude
loop
(4.60)
These
voltage
2v ,
J
and
(4.61)
[l/vQp )
(4.60^
resonan/
equations
FET's
1
steady
frequency
parameters,
not
by
a
the
Ipad
power,
resistor.
Th e
since
at
an
from
the
t wo
load
resistor
does
angular
frequency
FETs
in-phase
not
of
2
v
absorb
the
any
output
4
voltages
The
will
fourth
determine
function
equation
resonant
of
the
the
final
using
the
forcing
s a me
a
equation,
generated
input
.
the
driving
Bubharmonlc
voltage.
approach
as
voltage
We w i l l
wa s
equation,
as
solve
applied
to
a
this
the
ljfrtfp e q u a t i o n .
Rewriting
2X -
and
are
equation
(4.32):
2 ? M 2 - Q ^ ( a 2 L g mO $ + 2 ( a 3 Lo) ) R + ( 2 / Q
)Q
( 1 / Q „ )QQ + 2Cj a2Re ) Q + 2 ( <s3 R ) R + 2Q
P
®
”
S2 - S ? - ( l / Q
p
)SS
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 .
(4.62)
Letting
o . .J e j ( v T » n >
X -
^
- 2J X( o) ( a 2 e ‘ J
2
(4
l|' e :) 2
9
-a»
2
e-*
2e)
63)
.
1
S ”
-jS(o)(a-a*)
Q -
- 2 j Q ( o ) ( a 2 G” J
2
* - a * 2e i 2 'i>)
- 2 j R ( o ) ( o 2e"'J 2 V
Y -
X
is
t‘h e n o r m a l i z e d
is
isthe
the
is
the
is
v.
the
angmlar
of
function,
of
angular
3
forcing
varactor!
2
forcing
’
function
angular
output
voltage
frequency
v o l t a g e , ^ of
developed
2
v.
angular
v.
normalized
varactor,
Y
input
normalized
frequency
S
2
the normalized
a-cross
R
2*e^*)
- j Y( O ) ( a e ' J n - a * e 1 n )
frequency
Q
j *-a* V
of
subharmonic
angular
normalized
frequency
frequency
output
voltage
across
the
v.
subharmonic
voltage,
v.
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 prohibited w ith o u t p e r m is s io n .
of
-
Insertlng
these
values
4jX(o) ( a 2e " J
0
2
Into
'Pe J
+ 1 6 v 2j Q ( o ) ( a 2 e " J
20
(4.62),
-a*
a*
2
-
1 6 v 2 Q2 ( o ) ( o 2 e “ J 2 'f' + a*
-
1 6 v 2 Q 2 ( o
+
8
) (
a 2e _^
8 0-
2e
2e
2
26)
j 2 *)
e J 2 *')
^ - a * 2e ^
2
J
2
*^)
v ( a 2 Lu)p + ] / Q p ) Q ( o ) ( a 2 e ' J 2 ’t' + a * 2 e J 2 ^ )
(4.64)
a
+
8
v ( a 3 Lu> ) R ( o ) ( o 2 e ^ 2 ^ e ^ ^ + a * ^ e ^
P
4
+ 8v.1( 1 / Q p ) Q 2 ( o ) ( a 2 e _ j
2
'f' - a * 2 e J 2 ^ ) ( a 2 e “ J 2 ^ + a* 2 e j 2 ^)
-
2
,(/- a * 2 e j 2 *)
4 j ( l + a 2R ) Q ( o ) ( a 2 e “ j
8
- - 4 j ( a 3 R ) R ( o ) ( a 2e j
9 9
-
v
2
^e _J
a * 2e J
2
^e j * )
2
S
(
o
)
(
a
+
a *
)
A
-
v2 S2 ( o ) (
a
+
j
2
(
v
/
Q
Aga i n
be
.ignored,
equating
the
solutions
to
output
P
)
S
the
and
-
(
) 2
a *
o)
(
a
-
a
*
)
effects
(
a
+
a
due
*
)
to
the
expanded
V
coefficients
of
the
equation
and
resonant
the
*
loop
~
higher
driving
a
2
and
order
equation
a*
equation,
subharmonic
ha r mo n i c s
output
2
to
the
will
solved
by
zero.
Th e
0
fundamental *,
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 .
will
be
inserted
into
unknowns.
X( o)
and
equation
The; r e s u l t
Q(o),
where
>
‘- 4 j X ( o ) e ^
2
of
will
Q( o)
Rearranging
!c o e f f i c i e n t s
_(4.64)
to
reduce
produce
has
been
equation
S ( o)
the
as. a
previously
(4.64)
and
n u mb e r
f u n c t i o n , of
found.
solving
for
9* [ 8 v ( o 2 Lm + 1 / Q ) + 8 v ( a 3 Lw ) [ - ~ MH ] ■*•4 a 3 R [ - --H— ]' ] Q ( o )
P
P
P
1 + H2
8 1 +H2
2 v 2 S 2 ( o ) [ 1 - Kj / ( K flv Q p )]
( v j Q ^ S 2 ( o ) ( Kj / K 0 +l / ( v 0
/1 + C K j / K q )
/I +IKj/Kq)2
j[
8
t+ie
a2 ,
/l +(l/vQ
2
p )2
))
/ M C l / v Q ^
•
+
of
v ( a 3 Loj
) [ ——- —] P- 1 » H 2
(4.65)
4
( l + d 2 R ) ~ 4 ( d 3 R ) [ - —^ - ] + 1 6 v 2 ] Q ( o )
6
8
1 + H2
«
0*
►
*
^
+
. r ( v /Q
J
I
)S2 (o) t l - K
P
/ l + ( Kj /K o
»
) 2
1
1
/ ( K 0 vQ
)]
P
/ l + ( 1 / ( v Qp )
2v 2 S2 ( o )
[ K , /K 0 + 1 / ( vQ
_
) 2
1
/ l + ( K i /K q
y
) JP
/ l ♦ ( 1 / ( vQ
) 2
,
p
) )2 J
■* *
2
S i m i l a r l y one can d e r i v e t h e e q u a t i o n
for
t h e a*
\
’
1
T h i s - e q u a t i o n c a n b e s h o wn t o be t h e c o n j u g a t e o f
terms.
the
above
equation.
Combining
tion
equation
(4.65)
with
*
r e a r r a n g i n g the r e s u l t g iv e s:
and
1 6 X2 ( o )
*
[ J XQ( o ) - ( l ! +1 2 ) S 2 ( o ) ]® + ( j
2
its
conjugate
*
Q ( o ) > ( I 3 - I I, ) S 2 ( o )
ai jd
the
I's
and
J's
are
defined
] 2
( 4. 6~6)
%
where
equa-
below
(in
equation
(4.71).
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 prohibited w ith o u t p e r m is s io n .
(4.70)
n
Page
3.7
Monolithic
3.8'
FET M i x e r
3.9
Diode-ri.ng
3.10
Diode-Rect i f i e r
Phase
D e t e c t o r ..................................... ...
53
3.11
Single-Balanced
Diode
Mixer
......................................
55
3.12
Double-FET
D e t e c t o r ...................................................................
55
4.1
Field Effect T r a n s i s t o r Frequency Halver;
(a) P h y sica l c i r c u i t la y o u t ; (b) E q u i v a le n t
c i r c u i t l a y o u t . . . . ......................................................................................
61
Field Effect T r a n s i s t o r Frequency Halver
Nodal V o l t a g e and C u r r e n t I d e n t i f i c a t i o n . .
64
Boundary Region of u n c o n d i t i o n a l f r e q u e n c y
h a l v i n g as a f u n c t i o n of t h e s e r i e s r e s i s t a n c e . .
87
Boundary Region of u n c o n d i t i o n a l f r e q u e n c y
h a l v i n g as a f u n c t i o n of t h e g a t e t o s o u r c e
v o l t a g e . . . . . ...................................................................
87'
Boundary Region of u n c o n d i t i o n a l . f r e q u e n c y
h a l v i n g as a f u n c t i o n of t h e f e e d b a c k
c a p a c i t a n c e .................
88
V o l t a g e g a i n v e r s u s h a l v i n g f r e q u e n c y as
f u n c t i o n o f t h e l o q d r e s i s t a n c e .......................
89
4.^
4.3
4.4
4.5
4.6
4.7
4.8
5.1
5.2
5.3
.i.-.
Microwave
B a i u n . . . . . . . . . : ..................................
50
C o n f i g u r a t i o n s ......................................................................
51
Phase
D e t e c t o r ...................................................................
Phase
53
a
”/
V ol t a g e g a i n v e r s u s h a l v i n g f r e q u e n c y as a
f u n c t i o n o f t h e g a t e t o s o u r c e v o l t a g e . ........................
89
V o l t a g e g a i n v e r s u s h a l v i n g f r e q u e n c y as a
f u n c t i o n o f t h e f e e d b a c k c a p a c i t a n c e .................................. ^
90
( a ) P h a s e D e te c to r ; (b) Phase
E q u i v a l e n t C i r c u i t . . , .................
93
Defector's
P h a s e D e t e c t o r Out ' put V o l t a g e s f o r V a r i o u s
B i a s i n g C o n d i t i o n s , , ................................................
P Phase
Detector
Sensitivity
Bias C o n d i t i o n s . . .
1 15
D i a g r a m . . . . . . . . . . ...............
119
Frequency
6.2
Fabricated
Frequency
H a l v e r . .................................................
119
6.3
Monolithic
Frequency
Divider
1?2
-
•
.
Circuit
114
6.1
........
Divider
to
* •
Circuit
Diagram....
........................................................................ .
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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 .
-
Rearranging
[ (I
3 -1 1 4
equation
) 2+(l r+I2
) 2
(4.66)
82-
gives:
J S *4 ( o ) + ( 2 J 2 Q ( o ) < I
S 2 ( o ) + ( J'1 Q ( o ) . 3 2 + ( J 2 Q ( o ) ) 2 - 1 6 X 2 ( o 5
Solving
this
quadratic
equation
3
-
-I u ) - 2 J
1
Q(.o)
(I
1
+I2) ] •
0
04.67)
yields:
A2
-B ±
- 4 AC
- --------- —---------------------------------------------2A
S2 ( o)
(-4
wh e r e
A =
(I
3
-1^ ) 2 + ( I i + I 2
B =
2J
2
Q( T ) ( I
C -
( J i Q ( t ) ) 2 +( J
>2
) - 2 J j Q( T ) ( I
3
2
1
+I2 )
(4.69)
Q( t ) ) 2 _ 16X2 ( t )
l
1
1
"
2v2 -
12
“
13
=v
Jj
J
i
v*{K1 /Kq +
{1
-
2v 2 • {K j /K
“
8
“
-
l / v Q p } / Q p. J 3
q
+
(4.70)
l/v Q p } /J 3
v ( a 2 Ltop + 1 / Q p ) +
8
3
/vKo Qp }/Qp J 3
=
2
J,
{ l - K 1 / v K o Qp } / J
8
v ( a 3 L a: ) F 2
/ l + ( Ki /K
1
O
i 2" •
■ 2 v ( a 2 L oap + 1 / Q
-
v a 3 Lcjp Fi
4(a
2
/ l + ( 1 / vQ
) +
Rg+ l )
p
r
+ 4 a 3 R8 F 2
- 4 a 3 R g Fi
+
16v2
(4.71)
T
2 v a i i D p L F 3 + 2 a i R g Fi,
(4.72)
Kj
■ 2 v 2 - 2 ( a 2 R8 + l )
+
2
v a 3 0 )p LF l4 -
2 a j Rg F 3
F x - ^( 1 - MH) / ( 1 +H 2 )
^
s '
F*2
“
(M+ H ) / ( l + H 2 )
f 3 -
(1- p l ) / d t p 2 )
F *4 -
(L +P ) / ( 1 + P 2 )
(4 . 73)
*
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
-
83-
M “ <12 / 2 Yv
H ■ a3/2yv
(4.74)
j
P ■ a j / yv
L ■ a 2 /yv
We
have
nonlinear
input
signal,
pumpi ng
In
Y(o),
solved
differential
subharraonic
the
"now
has
four
equations.
S(o)
has
been
signal
and
the
equation
been
the
(4.46),
derived
the
in
terms
In
frequency-halving
equation
evaluated
FET' s
as
a
the
f u n c t i o n ( of
characteristics.
subharmonic
of
(4.68)
the
output
signal,
subharmonic
signal,
S ( o) .
Th6
•
1
voltage
next
required
section
4 .4
Analytical
plotted
a nd
this
initiate
a
Th e
will
be
boundary
t h,e ( v , X ) p l a n e
a
•
( 4 . 6 8 ) becomes:
is
of
.
subharraonic
the
the
caTled
of
the
foqnd
amo u n t
of
pumpi ng
.
generetiop.
by
Steady
the
State
pumpi ng
This
(v,X)
region
setting
voltage
pumpi ng
normalized
the
Response
required
voltage
angular
t
-B
be
frequency
v,
halving
on
plane.
of
S(o)
frequency
*= 0 .
Thus
/ b 2 -4A C
^ ----------
0 -
f
to
will
equation
^
*'
S 2 <o> -
to
division.
function
plot
the
_
now e v a l u a t e
frequency
as
evaluate
— V
Solution
We w i l l
initiate
'
to
*
will
*
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 .
-
Because:
A
84-'
0
then:
B o r:
/b
4AC *
Now,
2
-
4 AC
*
0
»
since
A is
A"-
greater'than
zero,
C mu s t
c
be
equal
t o zero.
Therefore,
from E q u a t i o n
2
-
f
16
j
(4.75)
division
occurs.
equation
gives
in
the
(v
the
equation
, X)
that
not
X( o)
region
has
the
boundary
normalized
v a l u e of
^ill
X(o);
It
follows,
a
frequency
frequency,
below
occur.
where
this
can
8
also
v
thiB
value
sub-
be
deduced
e m i - p a r a bo 1 i c
-
shape
„
been w r i t t e n
ofu n c o n d i t i o n a l
to
ca l c i / l a t e
frequency
and
plot
division.
the
As
a
«
obsefved
in
equation
boundary
field
-iA.75)
..
)
>
this
)
-— -
+ ( v/ Q
defines
plane. 1
A program
boun'dary
2
-
+K!
«
Given a
generation
this
wd* h a v e :
^
Equation
from
2
)-(K
------------— ---------------- Z
v
harmonic
2
4 ( J 1 +J2
XZ( o ) \ ‘
(4.69)
effect
is
%
equations
(4.75),
using
s
verydependent
on
the
(4.71,
parameters
4.72),
9
of
the
transistor.
*
Figure
as"
the
field
altered. .
graph
Th e
gives
simulate
these
4.3
s hows
effect
ta^le
the
curves.
J
effect
oft
transistor's
in
FET
the
-the
upper
series
left-hand
characteristics
.
this
which
boundary
region
resistance
corner
were
of t h e
used
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 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 .
is
to
-
In
the
resistapce
As
can
on
be
(v,X)
the
85-
plane,
the
subharmonic
seen
in
this
threshold
figure,
f
resistance,
Jhe greater
>
frequency-halving.
input
F i g u r e «4. A s h o w s
voltage
on
the
coordinate
is
pumpi ng
voltage
normalized
used
mo s t
In
that
for
subharraonic
to
other
'a
the
region.
generation
voltage.
*
Although
of
^
can
voltage,
_
the
required
»
is
to
the'gate
Here
(ratrher
<I t
occurs*
series
i s^ o b s e r v e d .
la/rger
gate-to-source
Input
the
series
»
voltage
figures).
given
level
the
voltage
the
of
«
influence
boundary
the
influence
to
the
bias
be
source
vertical
than'-.tjie
the
pUm- pi ng
voltage,
readily
on
.
X as
observed
bandwidth
dependent
,
initiate
the
>
*
be n o t e d
where
bias
\
*
here, it should
that
*
the g a t t - t o - s o u r c e
bias .voltage
alters
the point
a t ' whi ch, ,
* "■
* *
>
mi n i mu m t h r e s h o l d v o l t a g e w i l l o c c u r .
This occurs because
the
normalizing
In
ance
can
only
a small
(A.6
8
)
and
A. 5 t h e
observed.
effect
A second
frequency,
apparent
frequency* is
Figure
be
frequency
not
a
the
of
FET' s
has
been
gain
driving
t he * f e e d b a c k
feedback
subharmonic
voltage
fixed
dependent .
influence
Th e
program
d i v i d e r ’s
for
on
bias
as
a
capacitance
threshold
developed
to
function
voltage;
capacit­
has
curve.
c a l c u l a t e ,thf
of
using
the
input
equations
( A . 46).
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-
•»
In
Figure ,4.6
the
voltage
gain
as
function
of
the
halving
f r e q u e n c y i s o b s e r v e d , w i t h t h e l o a d r e s i s t a n c e as a
if
f
'
parameter.
T h i s b e h a v i o u r c a n b e * e a S ii l y p r e d i c t e d s i n c e t h e
“iiS
^
y
1
s m a l l e r t h e l o a d ^ r e s i s t a n c e t h e s m a l l e t w i l l be t h e v o l t a g e
C
gain.
(
I *
In F i g u r e
the
4.7
voltage
on
voltage
significantly
generation
voltage
octurs.
'
gain
at
voltage
the
influence
gain
is
alters
Al so
of
N
Th e
tJie/' oandwidth
the
\bias
^
gate^to
the
observed.
source
gate-to-aource
where
voltage
subharraonic
Increases
the
t
the
expe\ase’
frequency-halving bandwidth.
This
«
*
/
t h e l a r g e r ' the g a t e - t o - s o u r o e b i a s
capacitance.
„
He n c e
t hl e
of
*. d e c r e a s i n g
the
behaviour occurs because
* '
f
v o l t a g e , t^he s m a l l e r ’ i t s
capacitor
supports
a
greater
i
fraction
of
the
input
voltage
across
inductor.
then
-
Finally
feedback,
it
the
voltage
capacitance
is
in
input
^
gain
s h o wn
the
as
a
function
Figure
4.8.
of
The
the
small
*
influence
♦ ,
of
the
feed-back
capacitance
can' be
observed.
„
**
An
pumpi ng
i s . s hown
approximate
voltage
to
expression
input
power
for
(Pi n )
the
*s
conversion
easily
derived
from
and
below.
t
n
P IN -
4 v
[R
8
2
R,
E____ ?____________________ _
+ 4(v . v.L)
p
v
( l-l/(2v)
(4.76)
)
]
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
\
s u B H A M fo w q
th resh o ld
NO
R . . 10 D
0i
1.0
NORMALIZED FREQUENCY (-!
F i g u r e A . 3.
halving
Boundary r e g i o n
as a f u n c t i o n of
su bha rm
of u n c o n d i t i o n a l f r e q u e n c y
the a e r i e s r e s i s t a n c e .
Ow
ic t h r e s h o l d
L 1 0 n H
*. o r •
—
*.»*•*
* mi —
R , ' 0 B.
C . *00 p f —
C f 1.0 FF
IS
u
W O R M A LI Z I P F R E Q U E N C Y M
Figure 4.4.
h e l v i n g ea
i
,4
^ —1
. . .
‘---------------
p
■
*
b o u n d a r y r e g i o n o f ’u n c o n d i t i o n a l f r e q u e n c y
f u n c t i o n of t h e g a t e t o s o u r c e volt age".
"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 .
-
88-
»U>HAMtOM»C
thm hhold
to o
L H n H
•. 0 t •
v„ 1 0 .
S. it mt
s
», i 5mS
8 t ’0 » S
c. s o e p f
%
O
S
s
£
o
N
«3
*
m
o
T
C, JO
FF
1. . y
C, 10 *F
12
01
N O A M A U 2CO *A£OU€NCY
F i g u r e 4.5
Bo u n d a r y r e g i o n of
h a l v i n g as a f u n c t i o n of t h e
unconditional frequency
feedback c a p a citan ce.
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 .
89
PMOWCNCV -
HALVE* OAIN
10 mS
C . KM pF
C , t.O FF
0*
NONMALIZCO FREQUENCY (.)
Figure
V o l t a g e ga
f u n c t i o n of
4
n versus halving f
he l o a d r e s i s t a n c e
FREQUENCY -
HALVE X. OAIN
WO
L 1 0 flH
* ,
u
o r
>
37 mS
o, r» "4
■i S ">*
", 10 o
1.0 FF
1J0
NORMALIZED FREQUENCY w
F l g u re
.7.
Voltage gain
fu n c tio n of the
versus
halvin
ource v o lta g 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 .
- 9 0-
*
mCOUCMCV - H A lV lfrO A iN
*00
I 1 0 r>H
VOI TAQC
/
O AIN
\
C, 01
FF
Cf 10 FF
00
10
M OAM ALinD MCOUCNCV M
Figure
4.8.
Voltage
f u n c t i o n of
gi i i t i v e r s u s h a l v i n g f r e q u e n c y
the feedback capacitance.
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
CHAPTER 5
PHASE
DETECTOR
#*
The
requirements
frequency
halver
frequency
halver
I
where a
chapter,
for
have
wa s
realizing
been
outlined
theoretically
detailed
a
phase
In
Chapter
analyzed
understanding
detector
of
in
3.
the
its
and
a
The
previous
operation
wa s
revealed.
An
3,
will
analysis
now
be
All
systems
reference
ty.
or
signals
are
detector
performed
the
detector
on
the
phase
detector
some
have
one
a
form
combined.
slgnais
of
when
will
the
the
proposed
in
Chapter
chapter.
require
is
produce
However
this
mu s t
nonlinearly
signals.
phase
which
differences
phase
in
detector
will
and
the
synthesizer
A. p h a s e
sums
of
form
phase
of
a
As
in
at
mixer,
a
a
which
mixer,
dc
of
differ
signal
a
the
two
phase
which
the
two
only
in
which
difference.
to
capabili­
In
frequencies
signals
produce
locking
d' e t e c t i n g
frequencies
two
of
is
are
input
» >
phase,
dependent
^
c
5.
1 Phase
P
Defector
*
The
phase
Hodel
d e t e c t o r - which
is
to
be
analyzed
has
been
s
Introduced
Fig.
in
5.1(a).
transistors
Chapter
It
(FETs)
3.
This
consists
which
are
novel
of
phase
two
'connected
detector
separate
by
Is
shown
field
attaching
the
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
effect
v
source
terminal
FET.
of
The
gates
of
signals
the
the
c o mmo n
each
This
other.
terminal
the
of
The
be
used
phase
t wo
the
detector
represent
the
non 1i n e a r i t i e s .
( b ). t h e
The
C
gs
and
the
GaAs
C
(1
drain-to-s'ource
I
(V
ds
,V
) -
gs
ds
4r>o
is
the
C
is
the
8
is
the
is
the
X
is
t he
»a
is
the
Isolates
the
detector
is
a
D. C.
will
be
detector
The
model
the
to
t wo
floating
s hown
V
V
ds
is
point.
thi f^
[ 72}
principal
s hown i n
has
been
nonlinear
Schot t k y - b a r r i e r
Fig.
used
to
effects
capacitance
capacitance
-
V
gs
-V
gs
modelled
and,
as:
. . . .
(5.1)
o
current
B( V
♦
is
T
is
modelled
by
[72]:
) 2 (1 + XV
) t a nh ( cxV )
ds
da
b u t 5I t
in
potential
source
is
on,
output.
the
gs
input
that
•
V
the
biased
wh e r e
VT
lower
current.
ga t e - t o - s ou r ce
(V ) gs
it
MESFET
gate-to-source
drain-to-source
is
of
separately
equivalentcircuit
Curtice
the
terminal
phase
FETp
phase
Th e
(a)
the
presented,
as
drain
arepresented
When
5.1(b).
are:
the
configuration
the
analysis
can
to
signals
FET's.
from
terminal
FET
t wo i n p u t
the
Through
upper
the
source
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 .
(5.2)
\
-
93
-
d•
PHASE DETECTOR
OUTPUT
—
V
e»i
( b )
Figure
5 . 1 s.
(a)
P hase'Detector;
(b)
Equivalent Circuit*
Phaee
Detector's
\
«
‘ ... .
~j>
(
1
**
"
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5 . 2 Formulation
To
the
generally
the
(5.2),
Figure
We
In
except
by
phase
Su^pyiing
of
the
capacitance
begin
tlje
yields
at
high
using
det
currents
p h a s e d e t e c t o r , we w i l l f
Cp I s n e g l i g i b l e .
» This
frequency,
the
or
"5
nonlinear
circuit
at
the
where
Cp ' c a n
equations
(5.1)
representation
common
node
In
in
Figure
^
j
Vj +V
C( 1 -
analysis
case
511(b).
5.1(b)
the
feedback
b e ♦ 1 gn1f 1c a n t .
and
P i f f e r e p t ia-1 E q u a t i o n
simplify
assume ' t h a t
is
of
1-
r.*
S
V 3 -V
3
— 3---- ---------------- )
*o
*=£v
.*d/dt
^
(
i “ V 3 -*V 3 )
V ] , + V
© tr/-
+ S ( v 1+ V 1- u 3- V 3- V t ) 2 ( l + X( Vd s - v 3- V 3 ) ) t a n h ( a ( V d 8 - v 3- V
**
*
'
-
B( v 2 + V 2 - V t ) 2 ( l + V
’
l / R L ’ ( v 3+ V 3 )
v 3+ V 3 ) ) t a n h ( a (
3
))
v 3+ V 3 ) )
_
(5.3)
_
v
wh e r e
v1,
are
the
sinus&idal
^l»2»3f.are
the
direct leurrent
signals .
be
as:
The, i n p u t
2 , 3
signals
wili
small
represented
signals
v 1 - j v 1| s i n ( a ) t )
,
v 2 * | v 2 | 5 i n ( u ) t + 4>)
and
ignoring
assumed
higher
to
be:
v 3
“
| v 3a
(sin
(5.4)
order
a>t
harmonics,
+ |v
3
the
output
c |cosu)t
signal
will
be
(5.5)
•
*
To
normalize
simplify
the
voltage
<
•
the
analyslis
terms
as
well
of
as
equation
the
-(5.3),
coefficients.'
t
^
'
sJ
.jL:
* 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 .
we
will
-
95
-
/
Normalizing
provides:
Xb -
V i/* 0.
X -
za “
V3 / ' *o»
Z8 "
K "
g
T
<J»
/ R
Yo
.
,
X
L’ *
V2 / * 0 ,
“o "
“V
' Co
"
6
O
( Y ' - K ) 2 (l+X
where
(
Y'
* Y +Ys^n(wt+*)
equation
z
T.he
normalized
the
Th e
Me t h o d
{73],
obtained
Z ' ) - g TZ '
L
•
-
1
].tanh( a
o
/
V
)
(D-Z'J)
0
(5.7)
,
1
‘
^
1
(5.8)
phase
equations
(5.7)
and
(5.8)
define
the
detector.
defined
the
for
the
s-olution
for
phase
each
detector
terra
in
the
is
solved
equation,
we
differential
Z'.
differential
whereby
u al«n g t h e
nonlinear
equation
the
(5.7)
.coefficients
following
of
the
Z'
using
the
equation
Ritz
are
relations:
«
2
6
Solution
extract
(5.7)
.
+Z s i n ( u o t ) + Z c o s ( ait )
o s
c
Z'
Analytical
now
o
X + X s i n ( wt )
o
Having
can
Z')tanh(a
-
of
5
gives:
x '
behaviour
5.3
o
,
C»o
e ( 1 - X ' + Z 1 ) ~ * * d / d t ( X ' - Z ’ ) + S ( X ' - Z ’ - K ) 2 ( 1 + X f D- Z
o
o
o
-
| v2| / ♦ q
*
ro
(5.3)
Y -
l V3c l / * o ’ D " Vd s / *o
- X<| >
o
Rewriting .equation
Yq -
I V3S I / * o ’ ZC “
$o " B* o 2 ’
W
-
®L
| v i | / <0 o ,
x
C(Z• ,X',Y*)d( u t )
-
0
-
(5.9)
v
------
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-
2
tt
/
*
£(Z ’ , X ' , Y ’ ) s i n ( u > t ) d ( w t )
96 -
-
0
(5.10)
2 IT
/
o
C(Z, ,X ',Y l)cos(wt)d(a)t)
where
((Z
In
from
the
the
output
' ,Y ' )
above,
coefficient
the
Th e
obtained
the
dc
Ritz
of
the
te rr f i
odd
function
£,
separates
of
the
dc c o m p o n e n t
calculates
ande q u a t i o n
the
(5.11)
the
even
dc
Ritz
equation,
hand
side
of
the
left
the
(5.10)
function,
called
Defining
(5.11)
Equation
coefficient
output
0
(5.7)
(5.9)
signals,
output
first.
is 'equation
Equation
sinusoidal
calculates
as
1 ,X
-
function.
will
equation
be
(5.7)
gives :
f ( Z ' , X ' , Y ' ) « C *( 1 - X ' + Z ' ) ~ * * d / d t ( X ' - Z ' )
¥
+ 6
-
6
of
Th e
first
to
the
Ritz
term
of
(
X
'- Z
' -
K
)
2( +
1
X
o
[
D- Z ' ) ) v t a n h (
a
o
[
D- Z
'
J
)
)
(5.1 2)
o ( Y * - K) ‘ ( 1 + XQZ ’ ) * t a n h ( a o Z \ ) - g LZ '
The i n t e g r a l
side
o
for
each
of
the
four
terms
function
(5.12),
will
be
equation
(5.12)
can
be
on
the
right-hand
independently
e x p a n d e d and
give:
solved.
integrated
/
*
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 .
-
2u
1st
term
o
To
terms
of
function
binomial
we
will
use
the
out
TT
o i C
Solving
-
1
,
O
+Z
0
)
( z 6______c
-x)z
r
4 ( 1 -X + Z >
d(ut)
(5.15)
0 . 0
(5,16)
will
2nd
( x - z s )z c *
0
now
we d e f i n e
„
we h a v e :
yields:
Thus,
We
terms
4<1-X
/ l - x O +z O
three
(5.14)
periodic
2
_
first
approximation,
( i +x) ~* - i-±x+|-x 2
Integrating
(5.1 3 )
( 1 -X + Z + ( Z - X ) s i n u ) t + Z c o s a j t ]
0
0 .
8
*
c
simplify
the
-
u>C [ ( X—Z ) c o s c o t + Z s i n u t ] d( a j t )
o_______ s
c
/
■ I
97
Te r m
the
first
solve
the
term
second
of
term
dc
of
Ritz
the^
equation
dc
Ritz
equal's'
zero.
equation,
which
as:
^
z
°
where :
2 71
o
B ( X - Z -K + ( X- Z
o
o
o
X Z coswt)
o c
•
6
'
)sinwt-Z
tanh(a
o
coswt)
2
( 1+X D- X Z -X Z s i n m t
o
o o
o s
D - a Z - a Z s i n a ) t - a Z c os m t ) d ( y t )
o o o s
o c
(5.17)
r
G
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 proh ibited w ith o u t p e r m is s io n .
-
. For
Che
tanh
D- Z
and
function
o
we w i l l
98
-
as s ume
Z s i *nt ot +Z c o s tot
s
c
p
use
t anhX
1
-e
1
+e
(5.18)
- 2X
- 2X
<J
Hence,
expanding
equation
(5.17)
we
Cl + * 0 D - r Z o ) | #
have,
( X<>- Z (>- K ) 2+ i
0
80
( X - Z . > 2 . 1 B0 ZC 2 ]
*
r
■
I
2
Zo
,
-2a
■ 1 - e _____
.. -2a
1 +e
( D- Z
o______o J
( D- Z ) o
o
o
z
- X
[iB
O S l
O
(X - Z
O
O
- K ) ( X- Z
- X 2 [ - i B ( X ‘- Z
o c
o o o
Integrating
T
12
Z
this
equation
- 2 a ( D- Z )
i
°
°
1 ~ e______________
- 2 a ( D- Z )
o
o
1 +e
°“
8
) + j 0 (X - Z
0
0
- K ) Z - i s (X -Z
c
o o o
0
- K ) ( X- Z
- K) Z
B
)]
1
c
d ( wt )
(5.19)
gives:
(i +x d-x z
O
0 0
) [ e O( xO
- z -k) 2+ f e ( x - z )
O
O
S
2+i e O'z C2 ]
* 2Ott
-X
e Z (X - Z - K K X - Z
O O S O O
8
) + X 6 Z (X - z
o o c o o
- K)
(5.20)
This
We
c^n
equation
now d e f i n e
«
_rd
3
terra
A
- I
Z
3
the
defines
third
the
term
secon-d
of
the
dc
term
Ritz
of
the
dc
equation
Ritz
equation.*
as:
o
C'
?
where:
Z
I
3
o
-
2t
r
...............2,
/ - B ( Y +Ys i n ( u)t + <f>) - K )
0 0
•tanh(a
o
( 1 + X Z +X Z s i n u t + A Z c o s w t )
0 0
o s
o c
[Z +Z s l n w t + Z c o s w t ] )
o s
c
d( u>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 .
(5.21)
-
Using
the
previously
We h a v e
mentioned
99
-J
-
approximation
for
the
tanh
function,
f-
Z
-2a Z
2w
_
o
.
o o • (
13
" 1~e
/
- 2 a *Z
o
,.
o o
I +e
0
o
[ ( 1 + A Z ) [ ( Y - K ) + * Y 2 ] + A Z [ ( Y - K) Yc o s « t >]
o o
o
o s
o
+ A Z (Y - K ) Ys irM> ] d ( lot )
o c
o
(5.22)
Int egratlng ,
Z
°--
-2a Z
1-e
° °
-2a Z
o o
1 +e
(1*A
z ) [ < Y - K ) 2 + i Y 2 ] + A Z (Y - K ) Y c o s $
o o
L o
0 8
0
2,So
+ A Z (Y - K ) Y s i n *
o c
o
(5.23)
This
Ritz
equation
Is
the
solution
,to
the
third
term
of
the
dc
equation.
*
Finally,
we
define
the
fourth
term
of
the
dc
Ritz
equation
as:
jt h
4 .
A
Z
_
,
o
Te rm - I *
w h e r e :*
Z
*
2if
" / “ g L [ ZQ+Zs s l n u t + Z ^ c o s ut ] d( tot )
4
o
(5.24)
Solving ;
Y
Z
I,
-
(5.25)
- 2 * g L Z0
%
^Thls
provides
solution
of
equation.
' A\
We
the
now
have
completed
fourth
term
of
the
dc
Ritz
.
the
Integration
of, the
four
terms
A
w h i c h c o m p r i s e t h e dc R i t z e q u a t i o n .
To c o m p l e t e t h e s o l u t i o n o f
'
<
t h e dc R l t s e q u a t i o n t h e f o u r - t e r m s a r e summed and e q u a t e * ! t o s e f o ,
- 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 .
this
concludes
Che
derivation
of
the
dc
the
odd
Rite ^equation.
Th e
•
0
odd-term
defining
equation
(5.10)
called
function
Rita
e q u a t i o n , w i l l ^ n o w be s o l v e d .
{
Rewriting (5.10):
2n
/
o
Following
a
£ ( Z ’ , X ' , Y ' ) sinwt
similar
d( t o t ) ■* 0
approach
as
for
the
dc
Ritz
equation,
we
define:
I 8t
Ter m = I ^
where :
C s i n u ) t [ ( X - Z )(iscos(ji)t + Z u i s i n u i t ]
o_______
i_____s__________ c_________
i
f ( 1 —X + Z ) + ( Z - X ) s i n uit +2 c 0 8 (111 1
L
O O
8
C
J
£
2u
/
o
Using
a binomial
1
•(1+X)” 3 -
s_
2u(iiC
/l-X
we
can
let:,
(as
before
.
(5.27)
o
o
iZ
+Z
o
,
eq.
(5.27)
5.14)
»
*
l -i X+-rX2
^
Integrating
Z
expansion,
d ( ui t )
we
.9
C 64
.
. ■
get:
z
»
(z -X)2
,
'
2,3
3
■
( l - X* +Z ) 2
o
o
♦
6*
<1-X0 ‘ z 0 )^-
( Z -X ) Z
- i . -------^
f l
j
.o
o
( 5 :. T9
k )
We
2
nd
Term
can
define
the
second
term
as:
9
A
Z
8
■ I-
where:
Z
IZ*-
2*
J
■o
0 (X +Xs i nu) t - Z -Z s i n u i t - Z
o
a
o
o
c o » w t - K ) 2 * ( 1 + A D- A Z
o o
- A Z s i n<i »t - A Z c 6 i u t ) t a n h a ( D- Z - Z " s i n u i t - Z c o s uit ) • i n ait d( wt )j
c *
o c
o
6
*
^ . c
(5.30)
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 .
Solving
provides :
• Z
1 2* “
-2a
( D- Z
)
°
-2a.(D-Z
.
1l a+e
. o - o
)
1-e
(X - Z
O p
* 2ir8o
- K ) ( X- Z
8
) ( l + X [ D- Z ]
O
O
X Z ■ f ( X - Z - K)
0 8
0
0
2
+|(X-Z
S
) 2+i Z
C
2]
4 ox o zB
s ( X - Z os ) 2 - 4o x ^ zo„ 2c z se +r
^ Xo cZ 2 ( X - Z os )
(5.31)
We c a r f ' n o w
.
,.
define
the
third
term
as:
A
Z
Te rm • I 3 8
3r
wh e r e :
Z
I?
2» ‘ ® ■ / - 0 ( Y ' - K ) 2 ( 1+ X Z ' ) t a n h a Z ' ^ i n u t
'
o
o
o
o
' .
Solving
this
equation,
similarly
d ( cot )
to
the
(5.32)
previous
derivations
yields :
Z
%
13
1-e
1 +e
Z
° °
-2a Z
o o
Y( Y - K ) ( 1 + X Z ) c o s 4>+| X Z . ( Y - K ) 2
O
O ,0
0 8
0
-2a
We d e f i n e
4t h
(-2tt0o )
(5.33)
the
+7 X Z Y 2 - 4 x Y 2 z c o s 2 $ - 4 y 2 X Z s 1 n 2
4 O 8
8 0
s
* 8
o c
T
■*
fo urth term
:
A
z
Term - I„ 1
whe r e :
l• uF
zs
2n
* */ “ R.L ( Z O +2 8 s i n w t + Z ^ c o s wt ) s 1 n tot
O
Solving
Z
Ik'
(5.34)
d(iot)
glves :
(5.35)
■g L , ” 2 i
The
equation
•quate
the
values
have
now
summation
for
been
of
the
four
derived.
the
four
terms
To
terms
of
the.,
complete
to
odd
function
this derivation
zero:
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 .
Ritz
we
,
We w i l l
,
even
function
into
four
now s o l v e
Ritz
terms
t he n* combi ned
to
e q u a t I o n t we
equation.
produce
define the
each
terip
approach
term
equation,
which
called
the
separate
the Ritz function
'
4 *'
e
s e p a r a t e l y . These terms are
«
to
'
that
for
as :•
the
_
odd f u n c t i o n
Ritz
, < * ' ■ '
•’
*
.
ia
we
solution.
first
.
Ritz
Again,
the
a similar
L t
T e r m - I ! qC
1s t
third
and i n t e g r a t e
Following
*
the
<
/
'
■
’*
'
•
wh e r e :
Z '
_ c
11 -
'
2 TT
<■
f
j
^
C c o s ( u t f ( X - Z ) u ) c o s wt + Z ois i i i wt ]
o —
L
s
c______ » 1 '
•°
[ ( 1 - X +Z ) + ( Z - X ) 8 i*n wt + Z c o s w t l *
L
o
o
s
c
, J
Using
4 I
..
the
binomial
expansion: *
l +x) * - l - j x +•
Yields
the
T
I 1 c*
2 * ujC0
7l-X
<*("0
* «
„
x2
(5.38)
soljution:
o
6
* ( x- V
+Z"
o
, 6
3
V X
2
9
••
Zc
2
+ 6 4 ( x - z 8 >( t o t ^ z ) + 6 Z ( x - z B ) ( i H r « )
o
o
o
O'.
.J
+
< V X>
.
J "
n ^ To T ov
\
Ue
( 5 . 37)
have
solved
can
now
for
the
first
•
•
t e r m j of
the
even
(5.39)
function
Ritz
equation.
We
function
Ritz
proceed
equation,
which
to
solve
we d e f i n e
the
second
term
as:
1
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 .
of
the
even
•
A
Z
_nd _
_
c
2
Ter m - I 2
w h e r e :•
2*.
/ fe ( X ’ - Z ’ - X ) 2 ( 1 + X [ D- Z 1 ] ) t a n h a [ D- Z ' } • c o s wt ] d ( wt )
10
O
O
‘
o
•
C -
I?
N
Solving:
Z
T
e
2
' 2a
,
'
o
( D- Z
o
~Z ( X - Z
c o o
)
~ e - 2 „ ( D - Z ) ' 2 ' So
^
°
°
11 +e
This
is
solution
equation.
equation
^
the
1
for
- K ) 2~ i x
8
the
second
0
Z ( X- Z
c
term
8
) 2- | x
8
of
th«
.
We c a n
now d e f i n e
0
Z
c
3
even
(5.41)
function
«
the
third
term
of
the
even
function
Ritz
as:
A
Term - I
3
*
- K ) ( 1 + X [ D- Z ] + i X Z Z ( X- Z )
o
o
0
8
c
s
- j X Z (X - Z
O C O O
Ritz
( 5 . AO)
Z
3
c
where :
Z1-3 °
2*
/
'
■
’
r
'
I
f - 8 ( Y ' - K ) 2 ( l + X Z ' ) t a n h ( a Z ' ) ‘ c e s u t d ( cut )
L o
,
o
o
- J
,
.
0
’S o l v i n g
(5.42)
<*»
5 . 42 y i e l d s :
r
T
Z
c
-2a
.
Z
(Y - K ) Y( 1 + X Z ) s i n $ + 4
o
0 0
8
0 0
13
-2'*„
1
(5.43)
+e
+i X Z ( Y - K )
0
The . s o l u t i o n
proceed
which
X Z Y2s i n 2 $
O 8
to
solve
we d e f i n e
the
to
the
a
fourth
0
0
third
2
+ i X_ Z Y 2- i X Z Y 2 Co s -2 $
o c
o c
t e r*.m * i s
given
in
(5.43,,).
We 4 n o w
^
of' , t h e e v e n f u n c t i o n R i t z e q u a t i o n ,
terpr
as:
'
V
t
'
•
» " 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 .
t
-
4
104
-
Term
wh e r e
Z
c
1 1*
2ir
■ /
- g . ( Z +Z s i n w t + Z c o s ait ) ‘ c o s wt
L O 8
C
This
yields:
d ( wt )
(5.44)
Z
lit
C -
- g L »Tt «Zc .
Equation
(5.45)
Ritz
equation.
function
terms
of
zero.
the
even
Addi ng
three
are
four
the
Ritz
coefficient,
and
integrals
4t h r e e
dc
output,
a
We now
proceed
to
the
and"equate
four
the
using
to
the
three
o u t p u t 's
•
Ritz
their
terms
in
expanded
(5.16),
form.
(5.20),
i1 8 expanded
form
Using
(5.23)
the
even
integrated
summation
method.
algebraic
are
norraalize'd
rewrite
the
ZQ ,
Z8
to
even
and
We h a v e
equations
and
normalized
Zc ,
odd
function
in
wh i ch
function
coefficient,
‘
the
three
evaluated
1
in
of
(5.46)
unknowns
output's
respectively.
combine
term
o
derivation
Th e
the
fourth
we h a v e
Zc
the
now
equation,
+ i*
completes
normalized
integrated
can
Ritz
ZC
unknowns.
the
• We
+ i 3
three
the
terms,
ZC
+ i 2
This
s hows
function
the
ZC
i i
reduced
( 5 . 45)
,«f
e q u a t i on- ' - ( 5 . 26 )
(5.25),
Ritz, equations
we
can
along
rewrite
with, the
this
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 .
four
equation
*
-2a
1-e
!♦«
o
( D- Z
o
(i+x
)
- 2 a ( D- Z )
o
o
-X
( 0x
Z 6
0 8
Ms
0
Z
o, o
1-e
“- 25 a Z
T~
o o
1+ e
i 2 l t S ort
o
(x
-z
o
- k ) 2 +*b
o
.
- z - K ) ( X- Z
0
(1 + X Z ) [ ( Y
O O O
-2a
-
d- x z
o o
o
8
)+X
s
Z B (X
O C O
- K ) 2 + i Y 2 ] +X Z (Y
0 8 0
)2 +*b
( x- z
p
-Z
O
»■ n
z2 )
o c
-K) Z
O
- K ) Yc o s
C
4
+ X z (Y -K ) Ye i n *
o c
o
T
(5.47)
2 ' 8 LZo ■ 0
Similarly
(5.29),
(5.31)
function
Ritz
and
~ + i"
equation
(5.35),
equation
2 thi)C
VT-x o
using
(Z
9
64
o
s
o
( D- Z
o
1+ e
-2a
o
( D- Z
o
)
rewrite
the
- X)
.
( 1 -X +Z )'
o
o
)
+
can
along
with
its
solution
four
to
terms
this
odM
as:
(X
-2a
we
(5.36)
• ( 2 ir S ) •
o
-Z
o o
64
(1-X
-K ) ( X- Z
■i ( X - Z
o
s
) 2+ i z 2
c
+Z
o
3
32
)2
(x-z
(1-X
s
o
) 2Z
c
+Z
o
) 2
) ( 1 + X ( D- Z ] ) - i X [ ( X
o
o
o o
Z
J3 - ~QX~ ~
8
o
8
-Z
o
-K)
2
( X - Z.. ,) 2 +4"
v..
0 X Z z*
s
8
o s c
1
+ f X Z 2 ‘( X - Z )
4
o c
s
T.
(Y
-2a
Z
o o
-2a
Z
1-e
■•(
-
* L * Z.
o
’
2*8
o
- K ) Y ( 1 + X - Z ) c o s 4 + i X- Z (Y - K ) 2 + i X Z Y2
O
0-0
. 0 8 0
OS
) •
1 YX Z c o s 24 +4-Y2 X Z s i n 2 4
8
o s
Y 8
o o
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 .
(5.48)
-
Similarly,
Ritz
equation
we
cap
(5.46),
106
expand
using
-
the
the
solution
: t e r ms
to
(5.39),
the
even
(5.41),
function
(5.43)
and
(5.45).
V
i
Z
“X
Q
i ( x - ,z s ) + 644r ( x - z s ) ( t1 - -„-8■ ) -*-7r ( x - z s )(
X +Z
64
o
2 TT10C
/1-X
V z “o
o
c +Z
o
)
o
Z
+ 6 4 Zc ( l - X
-2a
o
Z
1- X
( D- Z
o
Z
)
(X - Z - K ) ( l + X [ D - Z ] )
c o o .
o
o ,
+ 4f x o Z s Z c ( X - Z 8 )
1-e
+
°
°
- 2 a ( D- Z )
1+ e
°
+'Z ) ( 1- X + Z 5
o
o
o
-2a
1- e
1+ e
o
*2 l , e o
■ix o Z c ( Xo
(Y
Z .
o
ix
0
These
/
) 2 ~ 8x
z
3
o c
last
the
Z (Y
o c o
to
the
function
dc
(5.49)
three
Ritz
Ri ' t z
equations
equation,
equation.
,
hyperbolic
are
the
To
odd
the
expanded
function
simplify
these
Ritz
f orme
of
the
equation
and
equations
let
the
>
tangent
equation
- K) 2 ♦y- X Z Y 2^ - f1X Z Y ^ c o a 2 d ,
4 o c
4 o c
T
'
solution
Ritz
Z ( X tZ
c
s
2 ir B
- g » i r » Z “K
L
c
dc
-4-8 Xo
- K ) Y ( 1 + X’ Z, ) s i n<{ , +- X Z Y 2 s i n 2
Y 8 O 8
o- -o
_2ab Zo
even
-K)2
-Z
o
function,
(5.47)
then
a 0 Z0
reduces
assumed
to
be
large
to :’
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}.
' The
4
-
107
-
I " '
“ ♦ Xo D- >o V [ 9o <Xo - Zo - K > 2 + * 8o ( K ' z 8 > 2 +* 6<,Zc 2 ]
.
- 1 0 Z . 8 0 ( X 0 - Z 0 - K ) ( X - Z . ) + X0 Z C 2 8 0 ^ , < 0 - Z 0 - K ,
-0
O
( 1 + XZ
- V
„
O
) [(Y
W
O
- K ) 2+i Y2 ] - 6
X I
0
g,-z
- r b r ^ oz
K)Y81"*
Slmllarily
0
the
odd
o
(Y -K) Ycos*
O
8
,
04
(5.50)
- 0
function
[u)C / / l - X +Z J ( Z / 2 ) + 8 ( X - Z
O
O OJ
C—
o o o
,
Ritz
equation
reduces
to!
- K ) ( X- Z ) ( 1 + X [ D- Z ] )
S
. O
o
,
*■
*
-0
X (Z
0
/ 2 ) [ ( X - Z - K ) 2 + i ( X - Z * ) 2+ \ Z 2 l - l 0 X Z ( X - Z ) 2
1 O%O
S',
c J 8 o o s
8
0 / 8
.4 b X Z z
o o o s c
2 + ^ 0 X Z 2 ( X - Z ) - 8 ( Y - K ) Y ( L+ X Z )cos<)>
4 o o c
8
0 0
* 0 0
'*•
-*0
0
-48
BO
%
«
'
X Z (Y - K ) 2 - j e x z Y2 4 s Y 2 x Z c o s 2 $
0 8 0
Ao o s
8 o
o s
Y 2 x
Z s
O C
i
n
Also,
[d)C / / l - X
O
0
+Z
0
2
-
the
’
(5.51)
0
even
function
Ritz
] [ ( X- Z .) / 2 ] - 0 Z ( X - Z
Jt
<8
J
o c o o
equation
-KMl+X
o
reduces
[ D- Z
o
to
J)
i
+-T0 X Z^Z ( X - Z ) - i 0 X Z (X - Z - K ) 2- 4 b X Z ( X- Z ) 2
A o o s c
a
o o c
o
o
8 o o c
8
”
5 0 „X Z * - 0
<y - K ) Y ( 1 + X Z ) 8 i n $ —4 b X Z Y 2 s i n 2 4 >
B o o c o o
o o
8 o o s ,
B A Z ( Y - K ) 2- i - 0 X Z Y 2+ 4 b X Z Y^ t f os 2 $ o o c
o
4 o o c
4 o o c
.
•>
0
(5.52)
\
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
began
In
To
summarize
by
assuming
Ch e
Che
draln-Co-source
non 1 1 n e a r i c i e s ,
phase
detector
phase
In
currenC.
we
Che
Fr om
formulated
behaviour.
were
solution
nonllnearities
these
in
analytical
Two
applied,
and
contained
a
a'
of
equation*
equation
sinusoidal
with
phase
detector^
gate-to'-source
Che
an
Che
these
inputs
varacCor
which
describes
which
differed
we
assumed
and
described
which
inputs
we
an
the
only
output
it
signal
which
cosinusoidal
then
term.
applied
weighted
After
the
and
after
s o me
and
(5.52).
terra
plus
normalizing
Ritz
integral
dc
Method.
the
sinusoidal
phase
The
Ritz
Each
of
these
they
wer e
equations.
simplifications
a
term
detector
Me t h o d
e q u atCon
required
equations
formulated
and
in
a
we
three
were
solved,
(5.50),
(5.51)
/
The
three
equations
obtained
uging
the
Ritz
Me t h o d
have
three
A
unknowns
(Z0 ,
normalized
Zg ,
output
Z£ )
which
signal.
We
are
now
the
proceed
coefficients
to
solve
for
of
the
the
three
second
order
unknowns.
Using
terms
such
equations
equations
2
as
and
Zc
solve
(5.51)
2
,
Zs
for
,
and
Y
Zc a n d
2
(5.52)
2
and
X ,
0 ,
(Y - K ) Y •
o
(
neglecting
we
can
rearrange
these
ZB :
c o s 4>-2 ( 6 •/
Z - ( 2 3 /<u C ) / l - X +Z
c
o
o
o o
and
) /l-X
0
o
+Z
o
( X -Z
o
o
- K) s i n 4 >
1 +( 2 8 /t oC ) 2 ( 1 - X +Z ) ( X - Z - K ) 2
o
o
o
o
o
o
(5.53)
( X- Z
8
) - ( 28
O
/uC
O
/T-'X O +Z O
)
f(Y
1
O
- K ) Ys i n $ + Z (X - Z
C
O
O
-K)]
(5.54)
■*
*»
From
(5.53?
the
and *. ( 5 . 5 4 ) ,
and
solve
for
the
equatiop-:
ZQ.
evaluation
we
of
can I n s e r t
the
Zc and
these,,
values
A f t e r s ome m a t h e m a t i c a l
^
Zg
f
terras
into
ine q u a t i o n s
equation
manipulation
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 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 .
(5.50)
we d e r i v e
-
0 -
F( Z
o
109
-
) - K ( Z ) c o s [ 4>- 0 ]
o
o
(5.55)
r-
where :
F( Z ) o
( 1+A D ) ( B - K ) 2 - ( l + A Z ) ( i Y 2 + C ' 2 ) + i I A 2 ( 1 —B ) C ' 2 Y 2 ( J + 1 )
o o o
o o
o
o o
o o
g 7 *z
+
K( Z
6
o
\)
o
-
-
A
A2 ( 1- B ) ( B - K ) / ( 1 + J )
o o
o
o
A C ' XY * I * f V -
o
■ t a n -1
o
/l
( 1+J )
-
'
]
— — r— ^
tanna
Z
o o
/
+ J/C1 + 2 J ) 2
,,
(5.56)
( / X / ( 1 + 2J ) )
wh e r e :
A
o -
B
20
o /<dC o
-
o X o- Z
C' -
Y -K
o
o
o
«
D
-
J
-
o
I
o
D- Z
o
o A2 ( 1 -o B) ( B o - K ) 2
1+ A
-
D +2 A B - 2 A K
o
o
o o
Equation
the
normalized
(5.55)
dc
output
Equation
detector.
input
voltage
The
dc
be
solved
Z0 a s
describes
equation
is
output
a
function
the
the
iteratively
of
from
the
obtain
<J>.
behaviour
climax
voltage
of
to
of
all
phase
the
the
phase
previous
detector
is
*
a
function
s i g n a l s . The
change
can
voltage
(5.55)
This
derivations.
*\«.
calculated
as
o
for
a
next
of
the
section
g iv e n ' phase
phase
derives
difference
'
variation.
the
between
the
p e a k - t o - p e a k dc
"
*
^
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 .
two
output
<,
-
Numerical
Typical
'detector
Justification
\
parameters
' /
/
..
K=- 4 . 2 86 ,
. 084 ,
Th e
externally
1.0, Y = l . o ;
X -
Given
ma d e
in
the
applied
Approximations
\
the
FETs
used
in
the
can
8 “ . 004*14,
signals
XQ = - 1 . 1 , ‘ Y0 solve
monolithic
phase
are
-2.0,
the
92 - . 0 0 1 ,
d e f *i r i e d
D -
o0- 1. 4
as:
11,429
equation
for
|az|
ahd
Z0 .
ZQ * 3 . 1 1
these
variables
phase
detector
APPROXI MATI ON
for
'
Cc - 4 , 7-6E-1 3 ,
At F = 5 GHz , ’we
h Z j = .104 ,
-
.
'/
are:
1 10
we
can
now j u s t i f y
the
a p p r o x i m a t i on s
derivation.
1:
(Z - X )
sinwt
+ Z c o s mt
8 ________________________________ C____________
1 - X
where
(1+X)- ^ " 1
Using
the
Zc = 1 . 4 1
Therefore,
o
-
+ z
iX + 3 / 8
above
cos<|>,
o
X2
parameters
( X- Zs ) = 1.41
equation
we
can
siniji
(3.14)
solve
for
Zc ,
(X-Z8 ).
/
X = . 27
and
exact
This
(1+x)"^
*
.8873
approx.
l-ix+3/8X2 -
gives
error
an
of
.8922
0.52.
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
APPROXI MATI ON
-
2
Z s i nt o t
♦ Z c o 6u> t
X^
8 ____________________________ C _______________
1
D - Z
Fr om
the
previous
■ 1.41
j)c
Za m 1 -
co bQ
1.41
eintot
X This
the
value
si n$
3 . 3.1 9
D- Z,
on
derivation,
value
factdr
APPROXI MATI ON
in
is
for
an
the
e
Error
-
above
-
e
- 2a
-2a
o
|az|
v.alid,
is
but* t h e
overall
effect
negligible,
because
this
terra.
Zo
2a Z
o o
x
page
Z
o q
parameter,
e 2 a 0 Z0 m ^ 2
.1
and
ZQ a n d
exponential
1. + e
therefore,
marginally
. 34
3:
.
1 -
Using
2.82
3.319
1. 41 jcos(tot-ji)
8.319
approximation
predicted
i*fc a
-
a0
1 . 4 , , Z0
-
3.1
1 0 “ **
. 9996
0.04%
\
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 .
10 6
-
5.4
Optimization
of
Phase
112
Detector
-
Sensitivity
9
H' a vi ng
form
of
reduced
equation
the
(5.55),
p Mi s e
detector's
we' c a n
now
use
nonliinear
this
equation
equation
to
to
the
calculate
&
the
dc
voltage
deflection
investigations
into
this
for
a
phase
given
'detector's
typical^sinusoidal
characteristics
can
output
assume
phase
the
dc
difference
Differentiating
phase
voltage
of
difference.
behaviour
multiplier
to
be
a
Preliminary
and
phase
sinusoidal
recalling
the
detectors,
one
function
of
the
[32],
(5.55)
with
r e s ’p e c t
to
the
ghase
$ gives:
- s i n ( 4>+ 9 )
aZo / d * -
dr i cz
) 7 d £—
o
o
.
° - 57>
. whe r e :
<■
H( Z 0 )
If
it
is
-
F(Z0 ) / K ( Z 0 )
assumed
deflection
of
that
AZ,
and
the
N
dc
with
output
an
(5.58)
voltage
average
iff
value
ZQ ,
sinusoidal
then
we
with
can
a peak
write:
t
Z
- 'l
o
o
+ AZs i n ( 4>+0 i )
.
■' ( 5 . 5 9 )
♦
Tj j e r e f o r e :
(
■ | A2 |
Th e
using
to
-
| d Z o / d » | MA)( -
p e a k - t o *-pe a k
equation
(5.60).
(5.60)
phase
detector
Taking
the
variation
derivative
of
can
H( Z 0 )
be
evaluated
with
respect
ZQ p r o v i d e s :
1A Z I ~
I
t
-
dH ( Z ) / d Z o
o
K•d F / d Z
------ 1
F * d K/ d Z
^--------------^2
-
wh e r e :
dF
dZ
(5.61)
^
-
2 ( 1+ A D ) ( B - K)
o o o
-
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 .
.
*
I
+ i
C ^ 2 Ac2 Y2 I
•
[ 1+ 2 J
•
f
°
T
( 1 - B )'
* 2\
- TB
— _- f y
} - —
o
o
{l
2B - K - 1 + A ( l - B
* Ao Ao
113
>
(1+J)
Cl-B0 ) ( l + J ) j
(5.62)
)
2"
°~ ' j ■ 8 L / t a n h a o Zo
Si n e e :
f ( z Q)
o,
*
We, h a v ^ :
l4Zl
In
|AZ|,
■
' t n h r o
order
equation
to
■
obtain
(5.63)
can
t
the
be
ma x i mu m
written
phase
directly
-
vpltages
and
that
bias
the
Figures
FET
5.2
and
have
5.3.
characteristics,
Rewritting
detector S e n s itiv ity ,
as
a
function
on
On e
the
(Using
Pin
dBm. )
(5.6)
^
can
phase
the
observe
DC v o l t a g e
dn
the upper
FET's
gate
Yo
■ V2 / d ' 0 t
DC v o l t a g e
on
the lower
FET's
gate
Zo
■ V 3 / $ o , / DC v o l t a g e
at
t h e outoput
AZ
is
output
In
the
next
development
compare
the
in
next
chapter
and
experimental
MMI C f r e q u e n c y
voltage
influences
in
measured
F-ET
terminal
deflection
for
the
phase
varying
.
design,
the
bias
sensitivity
experimentally
■ Vj/ $ o»
peak o u t p u t
the
detector's
XQ
the
of
*
characteristics.
voltages
(5 -63>
c h a p t e r we
of
thet h e s i s ,
fabrication
data
will
with
of an
the
Investigate
we
MMI C
\
theoretical
the
will
describe'
the
phase
detector
and
p r e d i c t i o n s ’.
'a
performance
halver.
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 .
of
a
MI C
Also
and
-
114
-
4 3
YO = - 1
YO = 0
YO » - 2
4 1
3.9
O
I
oX
YO = - 3
3 8
3 7
" YO = - 4
3 6 -
3 5
3.4
-2
Figure
5.2.
Phase
Va r i Va u s
Detector
Biasing
Output
Voltages
tor
Conditions.
*
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 .
Phase
Detector
Conditions
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 proh ibited w ith o u t p e r m is s io n .
-
116
-
CHAPTER 6
’
V
CI RCUI T
«
_u
/
DESI GN AND LABORATORY MEASUREMENTS
A
In
phaj se
locked
possible
divider
for
Chapter
are
the
were
detector,
and
the
then
phase
on
frequency
technology.
phase
that,
detector
Novel
and
In
of
and
the
However ,
detector
the
o^f
divider
and
for
divid-er
is
5,
phase
frequency
is
to
fabricated
dimensional
be
and
experi­
and. the
re.latively
t h ' ^ —frrrTC'bti on i n g
only
and
derivations
fabrication
MMICs
circuit
can
the
frequency
4
theoretleal
frequency
due
a
\
understand
the
all
Chapters
Constraints
outlines
of
for
MMIC c o n fH g u r a t i o n b
freque'ncy
These
technoldgy
divider,
mentioned
phase
the
requirements
detection,
chapter
ful t ^y
the
devised.
requirements
Th e
to
novel
wa s
understood.
wer e
measurements
detector.
thus
least
performed
the
This
mental
It
respectively.
revealed
division
outlined
components,
circuit
analyses
we
ors c l 1 1-at o r .
PLO
each
3,
phase
n e w,
of
the
first
on
and
novel(
hybrid
restrictions,
fabricated
using
the^
frequency
halver
Hybrid
circuit
the
MMIC t e c h ­
nology .
In
examined
be
This
this
chapter,
first.
Bot h
investigated,
is
then
and
followed
the
the
by
Individual
the
phase
and
will
be.
the
MMIC w i l l
performances
revealed.
detector
study.
v
In
V
the
latter
part
conclusions,
and
discuss
compares
the
measured
with
of
how
this
the
performance
chapter,
we
theoretical
of
the
will
draw
anajysls
circuits.
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 .
6 . 1 Ffcqliency
Halver
o-
TWo' t y p e s
•
using
a. s o f t
other
fabricated-
lithic
FETa
to
frequency,
on
initiate
which
we
circuits
which
.frequency
is
of
provides' excellent
and
design
th*.
Also
output.
strategies
and
one
circuit,
the
we
designed
call
as'
a
The
b£g
obtaining
of
amplification
Mono­
parametric
the '.nonlinearity
of
the
The
designed,
■
Hybrid
which
dlvis ion.
division
loop
required.
a
were
utilize
F ETs i s t h e p o s s i b i l i t y
*
*
thereby
eliminating
the
need
FET's
call
wafer
using
f
wire
'
a- s i n g l e
Bqth
halvers
dividers
'
substrate
bircuit.
frequency
of
v
of
Ga As
advantage
conversion
if'
gain
further
"serai-unilateral”
property *
isolation
between
the
following
section
explains
performances
of
the
of
resonant
Hybrid
and
the
MMIC
*
frequency
halvers.
*
6.1.1
, Circuit
*
Three
parametric
should
input
a
for
circuit
but
*
high
Second,
ma x i mu m
frequency
should
halver.
reasonably
frequency,
resonant
output
objectives
frequency
have
* designed
the
Design
sought
First,
the
quality
the
isolation
should
be
from
the
imbalanced
unbalanced
with
designing
resonant
factor
resonant
in
( Q)
circuit
at
circuit
output;
half
.should
and
w1 t h ^ r e s p e c t
respect
to
a .
the
be
third,
to
the
the. ^ i n p u t
f requency.
_
{
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
J
/
Hybrid
The
presented
source
the
in
Fig.
inductors
of
circuit
at
input
loop
t wo
is
t r a n s c o nd uc t aVi c e ,
together
form
applied,
a
c o mmo n
with
with
by
short
the
series
the
and
in
phase,
capacitance
a#1
the
desired
gate
resonant
to
of
the
the
gates
Schottky
subharmonic.
subharmonic
out
of
Th e
phase
across
the
device
,
subharmonic
g ra ,
connected
realized
a
halver
in
together
FETs,
produces
This
frequency
connected
are
and
nonlinear
maintains
FETs.
are
the
frequency.
9
the
by
gates
the
input
Cg ,
FETs
line,
of
the
me t
inductors
signal
where
dLodes,
resonant
These
the
are
the
impedance
half
FETs
barrier
with
capacitance
Th e
the
L.
high
source
of
conditions
*
6.1.
Th e
configuration
lengths
to
above
Mi cr owave F r e q u e n c y H a l v e r
is
%
appear
amplified
at
the
by
output
terminals
o fthetw oF E T s.
c
Since
frequency
and
balun
s ignaI.
subharmonic
while
^
a
F ^
to
the
balun
signals
signals.
provides
also
is"
also
output
combination
Th e
fundamental
filter
there
is
is
used
to
designed
F j n / 2* t o
of
t ©
the
than
balun.
the
octave
by
the
FET, ' a
cancel
suppression
the
of
suppress-
combining-
less
Ii n c r e a s e d
m a t c h i. n g
J
terminals
while
For
feedthrough
-input
filter
this
unwanted
the
in-phase
out-of-phase
operation,
the
attenuating
F ^
'
1T
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
FET 1
Flt; 12 (- 9 0 )
FET 2
FREQUENCY DIVIDER
^
Figure
6.1.
BALUN
Frequency
*2ikJ
Divider
Circuit
Diagram
V
r
Figure
6.2.
Fabricated
Frequency
*»
H a l v e r
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 .
-
The
is
mechanism
inherently
Zi n ( u ) ,
"pumped"
varactor
in
to
approximate
Harrison
for
an
the
from
[74]
input
has
wer e
input
signal
wa s
is
assumed
here,
from
the
Penfield
ion
used
to
to
as
then
u np umpe d
modified
of
impedance
Th e
impedances
the
approx in
equations
generation
p u mp e d
equations
obtain
These
value
the
different
impedance
impedances.
frequency.
quite
order
subharmonic
Thus
impedances.
Rafu6e[75]
linear
pre
■
parametric
nonlinear.
Zi n ( 2 u )
small-signal
of
129
a
and
t h e non­
obtain
an
fujnction
of
matched
to the se
impedances,
a
It
characteristics,
icantly
altered
amplitude
signal
ouput
In
the
Fig.
Th e
are
sina 11 - s i g n a 1
the
FETs
not
signif­
values,
t h e pumpi ng s i g n a l
of
the
by
[69].
wer e
FET
the
Th e s m a l l
then,
used
for
6.2,
the
fabricated
has
been
divide a
FETs
to
frequency
design
a
10 GHz i n p u t
u s e d w e r e NEC7 1 0
halver
i s,_ d i s ­
frequency
halver
signal
with
a
over a
0.3
10%
pm l o n g ,
]ate.
utilizes
Monolithic
Microwave
monolithic
frequency
the
nonlinear
common-source
wer e
isolation
used
in
between
element,
rather
the
of
area
impedances
"of
balun.
Th e
Th e
F ETs
of
frequency
bandwidth.
t wo
their
objective
woul d
recessed
from
output
impedances
to
played.
the
variation
matching
that
that
observations
FETs
order
their
t^ian
gallium
Frequency
halver
g a t e - t o - source
to
produce
to
take
input
and
distributed
arsenide
used
*
Halver
described
capacitance
subharmonic
advantage
output
element
for
the
here
of
"also
Cg
of
oscillations.
the
gain
terminals.
matching
circuit.
r "
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 .
and
Lu mp e d
minimized
-
The
The
basic
series
circuit
resonator
121
-
diagram
supporting
can
be
seen
oscillations
in
Fig.
at
6.3.
f <) / 2
wa s
k
realized
^fci ng
inductors
L3 .
the
Cg
of
Oscillations
high
impedance
presented
also
acted
a
Matching
drain
by
the
load
at
ground
applied
through
one
using
The
a nd
divider
the
ion
total
chip'size
In
the
measurements
frequency
6.1.2
were
implantation
next
for
halvers
’F r e q u e n c y
-
in
Fig.
a
Halver
on
-be
frequency.
and
the
the
did
not
to_ RF -
applied
through
.
lines
as
monolithic
1 x 400
active
pm F E T s
layers
Ga Ls
f or med
sub.strate.
pm x 1 5 0 0
three
with
provided
used
similar
Two
500
C2 .
circuit
1^
the
input
were
pm-thick
approximately
section,
by
fabricated
6.4.
2 50
loop
components
wa s
and
The
characterizing
will
Rj
the
isolation
C3
bias
spiral
L4
Th e
fabricated
into
wa s
by
s t r a t e g y followed
s h o wn
circuit
Gate
divider.
is
the
Capacitors
t wo
These
matching
combination
design
frequency
.
the
to
Cj.
at
output
L^.
choke
b a s ic
the
of
ancl
performed
circuit.
terminal
Hybrid
in
that
and
confined
circuit
being
ensured
FETs
\ , 2
wa s
Th e
used
matching
resonant
frequency
by
o u t p Xi t
resistor
the
wer e
the
FETs
the
the
as
bias
both
mos t
performance
pm.
relevant
of
the
defines
the
introduced.
Measurements
4
\
1
Th e
operation
of
the
of
characteristic
a microwave
turn-on
threshold
which
frequency
power
mos t
readily
halver
level
is
versiis
the
measurement
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 .
This
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 o w n er. 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 .
■-
curve
occur
''t
at
the
defines
given
t.he
bandwidth
a n/ i n i mu m p o w e r
input
will
be
A second
frequency
Is
as
The
input
evident
halver
this
that
of
measurement,
frequency
level;
from
for
a demonstration
third
-
where
measurement,
microwave
useful
123
a
conversion
varying
of
possible
input
mismatch
swept
gain
power
advantage
the
will
measurement.
of
the
division
of
a
levels,
using
response
of
FETs.
of
the
r -
halver
using
a
spectrum
output
powe r
levdl
gain
analyzer,
for
a fixed
provides
the
variation
input
power,
but
In
varying
frequency.
setup
All
of
these, measurements
s hown
in
Fig.
can
be
performed
using
the
6.5
A.
SIGNAL
GENERATOR
ATTENUATOR
ISOLATOR
r~" ■
VARI ABLE
ATTENUATOR
DIRECTIONAL
COUPLER
POWER
METER
FREQUENCY
DI VI DER
DIRECTIONAL COUPLER
ATTENUATOR
SPECTRUM
ANALYZER
Figure
The
6.5
next
Frequency
D ivider
s e c t i o n ‘ compares
Measurement
Apparatus.
the .measured
hmlymrm n ith thmlr th e o r e tic a l
performance
p r«d lctioii«.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6.1.3
Comparison
of
Theoretical
Hybrid
The
division
sion
gain.
the
circuit
'design
Figure
Experimental
Frequency
fabricated
around
and
Halver
1
successfully
frequency
6.6
Results
s hows
demonstrated
together
the
with
region
of
conver­
frequency
*
division.
power
As
Pin
threshold
tor.
For „£his
design
-11
power
4.75
and
and
4.76
supplied
by
mi n i mu m
dBm.
the
the
mo r e
threshold
point
comparison
with
of
input
and
this
the
resona­
signal
5% h i g h e r
input
than
level
[58],
halver
s h o wn
is
input
Using
from
FET' s
the
frequency
i n c r e a s e s as t h e
equivalent
and
Q of
conventional
frequency
t h e o r e t i c a l >c u r v e
better
about
mi n i mu m
the
the
threshold
operation
manufacturer
t u r : -on
Th e
a
and
in
this
of
on
centered
of
division,
mi n i mu m
seen
above
analysis
dependent
was
As
a mi nl r aur a . . l e v e l
initiate
the
bandwidth
increases
theoretical
is
circuit,
dBm,
the
to
level
frequency.
halvers,
previously,
required
signal
power^vas
that
mentioned
the
equations
characteristics
in
Figure
6.6,
p r e d i c t e d t o be
wa s
adjusted
in
the
experimental
the
-10*6
frequency
so
curve
be
can
observed.
Figure
6.7
illustrates
the
conversion gain
of
the
A ma x i mu m
con­
U-
divider
as
version
gain
PjN
wa s
off
due
a
to
at
combination
the
of
the
desired
of
Pjh
a *-
6 dB wa s ^ a c h i e v e d
increased
Go od
observed
function
above
onset
of
3.0
large
suppression
t^he o u t p u t
resulted
output
of
in
signal
dBm,
Fjn
GHz.
for
l ow
the
gain
signal
of
the
10.5
input
gradually
As
rolled
effects.
fundamental
divider.
being
powers.
Th e
frequency
filter
wa s
and ' b a l u n
n o m i n a l l y 36 dB
Fi 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 .
below
-
Ub
-
3.0 nH
-4
v
- 1 . 0V
Cp * 33 FT
On, * 20 ms
gL « 20 n6
.29 pF
-1 2
10 0
Figure.
110
10.4
INPUT FREQUENCY (QHz)
6.6.
Region
of
Frequency
Division
8 .0 — i
8.0 —
O <0
UJ
2 0-
-5
INPUT PO W E R (dBm)
Figure
4
6.7.
Frequency
Divider's
10.5
Conversion
Ga i n
GHz.
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 .
at
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 .
An
1
halver
iis
example
shown
amplitude
input
of
of
in
Figure
Fjjsj/ 2
frequency
the swept
6.8.
for
is
frequency
swept
ov'er.a
.fabricated
division
characteristics
measured
2 0%
frequency
higher
than
to
inductors,
thin-film
for
innput
The
theoretically
the
power
equations-
in
of
levels
output
bandwidth
For
suc^i
as
650
MHz
•Jg
frequency
in F i g .
^the
as
of
6.10.
6 . ^ and-
is
quite
response
for
approximately
the- models
of .-the
spiral
a
threshold
8.0
d Bm.
of
offset
swept
theoret­
with
has
been
inaccuracies
possible*
and
on
for
where
the
drive
an
A
input
a
can
theory.
18
applications,
sufficient
dBm. '
agreement
dependent
adequate.
9".-8
The
experiment
is
be
Division'-
as-model
being
a
The
can
frequency
dBm a r e
6.10.
offset
phas e-lock.ed-loop
14
is
response
This
between
GHz
the
demonstrated
reasonable
well
as
dBm,
frequency
ocfr^rs
is
operation
1.2
its
than
4 show
the
bandwidth.
a n d - FETs.
value
discrepancy
b a n d w i d t h * of
typically
seen
the
level..
power
greater
losses
of 2 . 0
value.
capacitors
the
represents
halver
Figures
in
of
Halver
and"
predicted
results,
to
4 0 0 MHz
division
chapter
contribute
^nput
in
predicted
Mismatch
a
shown
levels
omitted.
level,
12 GHz
inaccuracies
e x p e ri^roen t a 1
Th e
‘
the
signal
Frequency
at
the
spectrum
monolithic
where
attributed
ical
/
are
The
an i n p u t
• Monolithic
Th e
frequency response
d-Bm
lower
bandwidth
relatively
signal
.
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 .
flat
can
be
.
Th e
occurred
version
ant
to
for
the
input
loss
loop.
improve
mi ni mum
was
conversion
signals
attributed
A better
the
hybrid
choice
conversion
loss
loss
greater
t f4
the
at
12
than
low
of
input
to
a
GHe wa s
13
Q of
dBm.
the
inductor
value
9 dB,
which
This
input
and
is
and
con­
reson­
FET
can
comparable
configuration.
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 .
-
4
129
-
20-|
• 0 - 0. 7V
Vg, “ l.« V S
18-
C f - 26 FF
St, * 30 B6
Ol * 20
Sq * 2.5ms
CggfO) f .840 pF
16-
12 -
10-
"S'
F i gu r e
6.9.
Threshold
'
'*■
'
»' ■
%
Figure
6,10.
Halver
Power R e q u i r e d
Frequency.
:X
X:
dl
for
Division
: i* .
f t* " ! *
1-
<£»
Output Spectrum
Signal.
9 *
for
^
*
Againgt
?
4*r
S we p t
17 dBm I n p u 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 .
- 1 3 0 -
6. 2
Phase
Detector
This
v
monolithix
Fig.
the
section
source
detector
Each
input .is
and- t h e
phase
the
development
‘
utilizing
t wo
presented
to
detector
FETs
the
output
of
is
a
as
shown
gate
the
simple,
of
,in
one
of
c o mmo n
drain/
here
u tiliz ­
term inal.
Circuit
6.2.1
'-*•
phase
6.11.
FETs
describes
The
Design^
monolithic
phase
detector
described
:y>,
es
the
nonlinear
FET
and
the
to
g a te - to - source
nonlinear
produce
the
channel
desired
capacitance,
conductance,
mixing.
F ETs
Cg8 ,
of
have
the
the
of
one
other
F ET,
advantages
of
V
gain,
as
well
element
are^
of
as
matching,
gallium
6.12.
Cj ,
Capacitor
C5 on
of
C2 _ s h u n t
Sim ilarly,
provide
a
L. 2
than
used
input
with
-the
FET1
to
their
is
the
capacitor
FET1
FET2
Ci,
with
is
matched
can
a
gate
on
dc
be
using
applied
is matched
used
minimizes
diagram
is
bias
lumped
the
circuit.
re q u ire d for
source.
inputs.
distributed,
for
gate
combination
a
between
equivalent circ u it
t>
The
Lj-shunt
biasing
rather
arsenide
A detailed
Figure
isolation
seen
in ’
a
series
through
v gl-
block
using
bias
Vg2
the
drain
allowing
a
series
through
of
L2 .
FET2
to
the
RF
/
dc
block
for
bias
Vp w h i l e
grounding
?
signal.
<v
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
-
r
As
Is
evident
from
Figure
inal
between
t ’he
t wo
FETs ' w i l l
upon
application
of
bias
pass
filter
required
at
6.12,
the
become
a
connecting
dc
term­
floating
output
<s
is
unwanted
harm
* onics.
pro.vides
this
Th e
fied
of
and
FET2
appeaf
will
linear
mi x
its
drain.
with
to
L
Vs .
voltage
will
3
,
Cfc,
gate
Th e
of
FET1
signal
signal
from
capacitor.
The
t wo
1
fabricated
on
into,
pm-thick
250
x
400
active
1 5 0 0 pm,
as
at
pm
GaAs
shown
the
used
formed
and
will
Figure
be
to
-........................ -
-
-
L
—
-
across its
/
resulting signal
non-
t wo
will
input
detector's
Vq u j .
in
Th e
this
ion
total
circuit
-
were
im plantation
chip
6.13.
:
ampli­
FET1,
>
—
C7 m
gate
.s *
*—
the
the
using
substrate.
in
all
Li t ,
inpdt
terminal
FETs
layers
low-
characteristics.
the
appear
A
attenuate
i n t e r mo d u*l a t i on
products
of
the
✓
After
low p a s s
filtering,
the
phase
Th e
5 00 p x
the
and
the
signals.
a
point
filtering
to
at
Vj
combination
signal
gate-to-sourca
include
output
input
this
Th e
low-pass
voltages
...........................
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
size
is
-
The*
mine
the
selection
dc
detector.
of
reference
For
bias
value
simplicity
132
of
-
voltages
at
the
Vjj
output
measurement,
and
of
V8 was
Vg
deter­
the
phase
grounded.
<•
\
\
Vi
■ O "
I
PHASE DETECTOR
OUTPUT
j
—
- ©
—
T
Figure
a
6.11.
Phase
Detector
Using
FETs
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 .
-
133
-
r
tb
RF o•OVQyT
LO
«V
t
♦
;
C
*
■-............in. — i '
' .......... — ..... ■---------
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 prohibited w ith o u t p e r m is s io n .
-
6.2.2
Phase
The
Detector
134
-
Measurements
characteristic
which
most
readily
defines
at
phase
dc
detector's
voltage
set-up
sion
plot.
used
line
d e t e-et o r
measure
records
The
output
this
the
the
bias
is
6.14
shifter
between
voltage.
•s. 0
to
Figure
phase
difference
voltmeter
performance
I
phase
che
characteristic.
used
to
LO
inputs,
and
corresponding
supply
voltage
d* f f e r e n c e
illustrates
is
RF
the
is
used
"to
sensitivity
experimental
Here,
control
change
t ‘he
phase
optimize
for
an
the
and
in
a
veraua
exten­
phas^e^
digital
detector
the
given
phase
phase
change.
iso la tor
a tt e n u a t o r
EXTENSION LLNt
phase
directional
DlRECTlONAl
COUPLER.
DEVICE U N D E R
Figure
6 . 1 4 . ^ Phast
SHIFTER
Detector
COUPLER
TE ST
Measurement
Apparatu#.
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 .
-
6,2.3
Comparison
Phase
^width
of
at
5.0
terminals
was
found
isolation
from
the
measured
better
return
than
dependent
Vi g u p e
tion
6.15
of
power s ,
£
p, ower
clange
open
the
variation
in
differenced
RF
approximately
135
the
band-
RF
dB,
to
LO
while
the
dB.
The
terminals
we r e ,
16
found
input
with
mV p e a k
(jAZ|
a
func-
Various
193)
input
an
output
-
very
levels.
as
For
detector
be
power
measured.
phase
to
voltage
traces
were
ma x i mu m
wa s
output
Three
conditions
the
10
MHz
the
exceeded
sensitivity
shows
input
*
voltage
into
an
circu it. '
monolithic
sensitivity
compare
their
the
to
phase
are
In
voltage
variations.
required.
using
traces
tracer.
low
of
These
GaAs
values
accurate
can
either
be
a
in
order
of
to
against
parameters
determined,
low
good
frequency
for
to
or
the
differ­
a
high
method.
%
frequency
a
demonstrated
H o we v e r " ,
measured
These
characterization
the
detector
predictions,
a c c u r a c i e s , , by
y frequency
phase
experimentally
theoretical
FET m o d e l
curve
d.etector
bias
800
from
and
Results
respectively.
and
The
ing
LO
c o n d i t ion
0 dBm,
was
the
bias
and
an
than
terminals
at
4 dB
over
greater
RF
losses
Experimental
Isolation
the
phase
of
The
be
to
and
measured
GHz.
LO
phase
on
was
to
8 dB a n d
Th e
-
Theoretical
detection
centered
135
approach,
FET
measured
are
the
dc
»
measured
curves
are
,
,
current
on
then
a
versus
translator
used
___
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
as
an
-
optimization
tion,
goal
SPit-CE,
GaAs
FET
Using
for
SPICE,
including
X,
comparison
of
the
operating
the
Rd <j
to
for
frequen cVfs
parameter
(X)*
RDS »
^
can
imately
MHz.
the
curve
It
A
as
by
is
primary
GaAs
FET
to
be
the
in
the
[77] .^
Rpg
approximately
on
10
at-
In
by
the
increasinverse,
1 MHz
wa s
percent
relatively
from
constant
1 MHz.
t,hat
X
to
then,
is
inversely
increase
frequency
assumed
of
a
current
curves
primarily controlled
appears
fact
the
the
dependent
affected
measurement
[72].
traces.
that
these
a
through
ind icate d that th is slope
v
f r e q u e n c y , and c o n s e q u e n t l y i t s
estimated,
10
of
of
parameters,
have
above
the
curve
very
slope
is
1^0 Hz .
Using
are
Th e
have-dropped
v a l u e ^ at
FET
demonstrated
curves
simula­
model
deduced
simulated
slope
[77].
its
Curtice
be
a^ep a r t i c u l a r l y
this
decreases
been
such
r e p r e a e n t a t i oo n
the
can
and
One
frequency
Vj
[76).
increasing
observed
has
volta t e
modulation
with
and
it
Measurements
es
from
low
measured
region
simulation,
channel
a
frequency
saturation
time-domain
v a r ioup
£3,
drain
a
-
simulations.
parameters
Recently,
versus
circuit
provides
using
136
is
af a c t o r
increased
for. high
parameter
-by
that
proportional
of
from
10
frequencies,
alters
the
traces.
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 .
to
approx­
Hz
to
that
X
1
is
simulated
y
i
-
137
J
Th e
is
shown
ir^
voltage
(
measured
Figure
and
phase
6.15
port
2
detector
as
a
output
function
gate
bias
voltage,
o f ? port
voltage.
v 0UT
>
1 gate
bias
Figure
5.3
(
illustrates
| AZ|
is
as
a
the
function
observed
reaches
a
that,
value
In
output
the
input.
The
a
sensitivity.
in
Figure
Vi
■
however
Th e
V2
the
The
observing
Figure
pinch-off
the
apparent.
sensitive
,|
■
The
to
an
for
output
It
| AZ |
measured r e s u l t s /
*
theoretical
prediction
for
a
given
curves
of
the
difference
at
been
normalized
to
the
output
is
phase
voltage
at
is
shown
P l N * ” 0 dBm,
detector
s h o ws
det^-ctor's
voltage
observed
phase
of
phase
have
average
agreement
-1.5V,
average
agreement.
, I
“
sensitivity
the
comparison
predicted
Good
to
the'
observed,
direct
4^15.
1.5V,
6.15,
theoretical
more
detector
g a t e b i a B v o l t a g e s , XQ a n d Y0 .
'fa*
given a p p r o p r i a t e bias (Conditions,
Figure
is
phase
of
comparable
voltage
provide
calculated
sensitivity,
considerable
dis­
associated
error
can
be
predicted
from
"j
>
0
5.3,
as
the
FET6
are
biased
closer
to
sensitivity
to
theoretically
accurate
value
bias
voltage
predicted
for
the
becomes
values
pinch-off
:-------------------:---------,-------------- *-----------,
are
more
very
voltage.
------
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
138
-
-
0
\
PHASE DETECTOR
3.250
’ 3.200
'1N - 0 (ll»n
'
\
3100
P,N x 0 dbm
- -0. 7V
out
2 = -2 1 V
cn*
5 3 000
o
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ou t
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2.900
2.800
2.750
2 700
F i g.u r e
6 ^1 5
120 ISO 180 210
PHASE (DEGREES)
. Me a s ur e d
Function
Phase D e t e c t o r
o f V o l t a g e and
240
270
300
330
C H a r a c t e r 1 s t i cn
Power.
o
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 .
as
a
6.3
Conclusions
This
the
section
freqqency
halvers
In
the
of
the
purpose
e x <Y
perimental
I 0
detector^
summarizes
case
and
phase
of
the
measurements
r e sXu l t s
the
hybrid
wa s
frequency
to
determine
for
halver,
the
the
region
of
f r e q u e n c y h a l v i n g as w e l l as t o d e m o n s t r a t e s u b h a r m o n i c con-j
•>
*
version
gain
using
Field
Effect
Transistors.
The
smajll
losses
uted
associated
to
the
performance
good
of
One
dBm.
the
FET
hybrid
the
by
compares
the
analytically
of
FET's
a
(ontrH>predicted
with
simple
calculation
characteristics,
threshold
well
halver
halver.
use
equivalent
turn-on,
frequency
with
frequency
show,
halver
This
the
a g r e fc^me n t
can
Incorporating
frequency
wi \ ^h
a
point
measured
that
the
should
be
-10.6
value
of
-11.0
9
dBm.
The' agreem ent
resonant
look's
subharmonic
parison
in
threshold
Th e
particularly
unmatched
conditions
frequencies.
Fig.
6.6,
boundary
hybrid
is
It
that
the
regions
frequency
can
also
the
also
considering
at
the
be
observed,
theoretical
follow
halver
good
fundamental
and
same
by
a
conversion
gain,
as
wa s
experimental
geometric
demonstrated
shown
in
Fig.
6 t 7.
/
ilim S M —
—
III!
IT
■
i ■/
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . I .. . . . . . . .
■. . . . . . . . . . .
and
com­
shape.
6
A
Subharmonic
the
,. . . . . . . . . . . . . : - - - - - - - - - • •
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
dB
-
The
monolithic
dBm.
to
halving.
loop
This
+8.0
of
freqency
frequency
resonant
purpose
d Bm.
have
the
these
laboratory
halve'V
wa s
to
Because
of
the
measured
slightly
higher
Th e
measured
threshold
same
general
low
effects
any
making
The
a-
deviation
than
near
the
as
output
step-function
diminishes
for
v.
point
wa s
the
laboratory
detecting
power
to
wa s
I
be
power
very
of
cases,
large
Figure
v<l,
also
dependent
inputs.
increasing
in
an
obseryed
optimally
k
__
on
the
increase
that
predicted
the
For
as.
bias
both
the
values,
wa s
also
'•
observed
the
input
i
p rin c ip a lpurpose
demonstrate
As
the
predicted
conditions
factor
as
of
phase
from
wa s
well
the
found
as
the
theoretical
power
its
tq
and f e x p e r i m e n t a 1
%
the
phase
detector
sensitivity
gate voltages,
the
sensitivity
factor.
-It
approached
factor
#
'
shows
gradually
ra y id ly .
■
signal
4.6,
which
sensitivity
input
in
the
circuit.
the
found
invalid.
behaviour
wa s t o
the
investigation,
the
resulted
of
of
theoretically
very
jurve-,
for
detector,
*
measurements
capability
theoretical
to
phase
+9.8
value
in the h y b rid fre q u e n c y h a l v e r ( F ig . 6 . 8 ) , however
.
m atching c i r c u i t r y tended to
ma s k t h i s o u t .
For ; t h e
of
subharmonic
region
assumptions
This
the
*
turn-on
larger
the
that
on
region
predicted
boundary
attributed
equation's
predicted
Q of
the
shape
being
its
threshold
4
derived;
measurements
determine
turn-on
is
the
T
1A 0 -
- -f
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
wa s
their
increased
^
-
1 4 1 -
*
In
summary,
at ion- p r o v i d e s
FET
frequency
the
insight
halver
frequency
into
and
the
the
divider
operation
analysis
can
theoretical
of
the
be
used
deri'v-
param etric
to
select
the
o p t i m u m FET a s w e l l a s I t s b i a s i n g c o n d i t i o n s . A l s o , t h e
I
p h a s e d e t e c t o r b e h a v i o u r has been t h e o r e t i c a l l y I n v e s t i g a t e d
In
detail,
phase
with
this
analysis
providing
Insight
d e te c to r's ’ operation.
V»
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 .
into
the
CHAPTER 7
" ACHIEVEMENT AND RECOMMENDATIONS FOR FURTHER
*
7.1
Achievement
■
The
RESEARCH
0
purpose
of
this
thesis
has
been
the
investiga.
tion
of
key
‘Following
that
the
and
which
divider
monolithic
a
preliminary
monolithic
To
the
the
phase
verify
most
which
t h e .concept- of
FET
halver
wa s u n d e r t a k e n . F u r t h e r m o r e ,
of
wa a
the
divider
observed,
*
relationship
harmonic
The
it
were
locked
loop.
was
determined
leasvt
understood
are
the
frequency
(based
of
frequency
a
division,
hybrid
f o r m *bf
a mathematical
an
the
deriva-
principal
on
the
c h a r a c te r is tic behaviour
• 1 I
FET's e q u i v a l e n t
parameters)
both1 e x p e r i m e n t a l l y
.
between
the 'turn-on
and
t h e o r e t i c a f l y* ,
threshold"'
with
in
the
£he
sub-
frequency.
Having
laid
4
the
foundations
for
the
FET
frequency
.
divider,
a
monolithic
tested.
^
Due t o
detector
hybrid
form.
and
necessity
V.
w
couvl d
of
c^ose p r o x im itie s ,
the
novel
V
n o t - be i n i t i a l l y
investigated
In a
^ "*, .
*
C o n s e q u e n t l y , a M a t h e m a t i c a l d e r i v a t i o n Of t h e
characteristics
design
v e r s i o n wa s , d e s i g r f e d , f a b t i c a t e d ,
t
!
•
the
phase
vl n l t l a l
investigation
developed.
phase
detector.
experimental
wa s
a
development
initial
tion
of
investigation,
components
required
and
components
of
tool.
this
novel)phase
Again,
detector
having*laid
the
served
as
the
foundations
for
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 .
t h i i v phase
ricated,
detector
and
In
the
monolithic
summary,
conceptual
the
halver
design
and
sim plicity,
measurement
version
was
designed , fab-
tested.
frequency
7.2
a
and
and
implementation
the
phase
with
good
detector
of
both
y i e l d e d ' 0 to
agreement .
between
expectation.
Recommendations
for
<■
Further
Research
*
The
following
could
be
considered
for
further
research:
1.
The
development
halver
by
with
using
output
2.
The
3.
An
monolithic
losses.
monolithic
of
the
into
of
all
a systea
on
into
components
Potentially,
the
microwave
This
could
am plifiers
frequency
investigation
passive
f
a
reduced
integration
nents
%
of
at
frequency
be
the
achieved
input
and
divider.
the
phase
gallium
producing
on .
locked
compo­
quality
factor
arsenide.
high
a
introduction
loop
monolithic
of
a
high
wafer.
dielectric
\
.
constant
reduce
{A.
The
the
the
onto.
the
wafer
of
a
integration
of
silicon
hybrid
GaAs-Silicon
integrated
t h e RF Ga As c i r c u i t r y .
< b
graphical
representation
for
A
* »
■r v a l i d i t y
could
greatly
losses.
development
nology
S.
m aterial
wafer
circuit
with
of
the
for
tech­
'
the
^region
as'sumpt i o n s .
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 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 .
of
-
144
-
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J.R.,
^
Las
. . . .
■-
■■
rejection
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...
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Ga As
U.S.A.,
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R.S.
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19.
H. M. ,
146
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’^ H y b r i d
Matter
Ho wa r d *
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27.
Rottersman,
*
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M.H.,
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and
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*
Te’c h .
D. ,
1,
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1978.
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1 37 .
* 1
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30.
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June
643,
the
and
Theory
and
1976.
1979.
development
of
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dralri^-s o u r c e
Microwave
312-317,
27,
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with
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