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# Full-wave analysis of nonlinear active microwave circuits in packaged environments

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U
n iv e r s it y
of
C
a l if o r n ia
Los Angeles
FULL-WAVE ANALYSIS OF NONLINEAR
ACTIVE MICROWAVE CIRCUITS IN
PACKAGED ENVIRONMENTS
A dissertation su b m itte d in p a rtia l satisfaction
of the requirem ents for the degree
Doctor o f P hilosophy in E lectrical Engineering
by
C hien-nan Kuo
1997
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UMI Number: 9737339
UMI Microform 9737339
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copying under Title 17, United States Code.
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The dissertatio n o f Chien-nan K uo is approved.
<xlc £-~
D i “(‘-S o n P;an
N athaniel Grossman
T a t s n o [to ll. C’o n u u ir te i* C'haii
U niversity o f C a lifornia, Los Angeles
1997
ii
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To m y fa m ily
iii
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T able of C o n ten ts
LIST OF F IG U R E S ......................................................................................
vii
LIST OF T A B L E S .........................................................................................
xii
A C K N O W L E D G M E N T S ............................................................................ xiii
VITA , PUBLICATIONS A N D P R E S E N T A T IO N S ............................xiv
A B ST R A C T OF T H E D ISSE R T A T IO N ................................................ xvii
1
2
IN T R O D U C T IO N ..................................................................................
1
1.1
Problem D e s c r ip t io n ...............................................................................
1
1.2
L ite ra tu re S u r v e y ......................................................................................
2
1.3
C o n tr ib u tio n s ............................................................................................
5
E X T E N D E D F D T D M ETHOD
.......................................................
9
2.1
Conventional F D T D A l g o r it h m ...........................................................
9
2.2
Im provem ent o f Num erical E ffic ie n c y .................................................
11
2.3
Incorporation o f M u lti-te rm in a l Active D e vices................................
13
2.4
3
2.3.1
Equivalent C urrent Sources
......................................................
15
2.3.2
E quivalent Voltage Sources.........................................................
17
D ifferential E q uation S o lv e r .................................................................
23
SM ALL-SIGNAL ANALYSIS OF AM PLIFIERS
.....................
iv
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25
4
5
6
7
3.1
I n tr o d u c tio n ..................................................................................................
25
3.2
M ic ro s trip Line Term inated in to a Matched R e s is to r........................
26
3.3
S im u la tio n o f a Microwave A m p l i f i e r ...................................................
28
3.4
E ffect o f M ism atching
..............................................................................
36
LAR G E-SIG NA L ANALYSIS OF A M P L I F I E R S .....................
40
4.1
I n tr o d u c t io n ..................................................................................................
40
4.2
Device C irc u it M o d e l.................................................................................
41
4.3
F D T D S im u la tio n ........................................................................................
44
IN V E STIG A TIO N OF PA C K A G IN G E F F E C T ........................
52
5.1
I n tr o d u c tio n .................................................................................................
52
5.2
S im u la tio n o f Packaging E f f e c t ................................................................
53
5.3
O s c illa t io n .....................................................................................................
58
CROSSTALK IN MULTICHIP M O D U L E S .................................
61
6.1
I n t r o d u c t io n .................................................................................................
61
6.2
M o d e lin g o f the M o d u l e ..........................................................................
63
6.3
C rosstalk Analysis
....................................................................................
65
6.4
R e duction o f Crosstalk B y S e p a ra to rs ..................................................
69
ABC REALIZATION B Y DIGITAL F IL T E R S ...........................
73
7.1
I n tr o d u c tio n .................................................................................................
73
7.2
T h e o re tica l F o r m u la tio n ..........................................................................
74
7.3
A b s o rp tio n o f Single Modes
78
...................................................................
v
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7.4
A bsorption o f M u lti-m o d e s .......................................................................
81
8 C O N C L U S IO N S .......................................................................................
84
R eferen ces..........................................................................................................
87
VI
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L i s t o f F ig u r e s
2.1
The
p o sitio n o f each field component in a F D T D cell.................
2.2
The
flow diagram o f the F D T D m ethod in each tim e
10
update field components..............................................................................
12
2.3 A packaged tran sistor connected to the g ro u n d plane through vias
at th e source p o rt in a m icro strip c irc u it................................................
14
2.4 The im p le m e n ta tio n o f device in co rp o ra tio n using equivalent cur­
rent sources.....................................................................................................
16
2.5
The
17
2.6
The flow diagram o f the F D T D m ethod in each tim e advance to
N orton-equivalent c irc u it o f device-wave in te ra ctio n ...........
update field components..............................................................................
2.7
The replacem ent o f the active device by equivalent voltage sources
on th e F D T D g rid edges..............................................................................
2.8
18
The equivalent circu it of two F D T D meshes on the both sides o f
a voltage source.............................................................................................
2.9
18
20
(a) T he equivalent circu it o f the co n fig u ra tio n in Fig. 2.7 as seen
from the device at each p o rt, and (b) th e Thevenin-equivalent
c irc u it o f device-wave in te ra ctio n ..............................................................
21
2.10 The flow diagram o f the F D T D m ethod in each tim e advance to
update field components..............................................................................
3.1
3.2
22
The side view o f the configuration o f a 5 0 -fi m ic ro s trip line te rm i­
nated w ith a 50-Q resistor...........................................................................
26
The re tu rn loss o f the m icro strip c irc u it in Fig. 3.1............................
27
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to
3.3
(a) The picture, and (b) the layout o f the microwave am plifier,
designed a t 6 GHz, for small-signal analysis.........................................
3.4
The device c irc u it model o f the packaged M E S F E T ...........
3.5
Snapshots o f field d istrib u tio n at d ifferent tim e steps, (a) 320, (b)
30
640, (c) 960, and (d) 1280. ( C ontinued) .................................
33
3.5
( C ontinued) ..................................................................................
34
3.6
The observed tim e responses at the in p u t/o u tp u t ports of the m i­
crowave a m p lifie r
3.7
35
The calculated results of two different approaches and measured
S-parameters, (a) I S 2 1 I , and (b) |5 n |........................................
3.8
29
37
T he calculated and measured S-parameters, (a) |S-.>i|, and (b) |S u|The numbers o f voltage sources used to represent the gain and
d ra in p o rts o f the device are chosen as param eters..............
4.1
39
Inside the dashed box is the large-signal device c irc u it model o f a
M E S F E T used in this paper. The gate-source capacitor Cgs and
the d ra in current source I is are no n lin e a r................................
41
4.2
D C characteristics o f the M E S F E T ..........................................
42
4.3
The stru ctu re o f a microwave a m plifier fo r large-signal analysis.
4.4
The config uration o f source e xcita tio n .....................................
45
4.5
A snapshot o f field d istrib u tio n d u rin g s im u la tio n ...............
46
4.6
The observed tim e response at the in p u t/o u tp u t p o rts ......
46
4.7
Sm all-signal analysis of the microwave a m p lifie r a p p lyin g the largesignal c irc u it m odel........................................................................
48
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.
44
4.8
4.9
The spectrum o f the o u tp u t power using single-tone e xcitatio n o f
different power levels a t 6 G H z.................................................................
49
The o u tp u t power o f harm onics using single-tone e xcita tio n .
50
. ..
4.10 The spectrum o f the o u tp u t power using tw o tones o f the same
power level Pin at 3 G H z and 6 GHz.......................................................
51
4.11 The o u tp u t power o f in te rm o d u la tio n p ro d u cts...................................
51
5.1
The structure and dim ensions of a packaged m icrowave am plifier.
The package is m odeled as a PEC box w ith two holes at the in ­
p u t/o u tp u t ports fo r feeding power..........................................................
54
5.2
The F D T D sim u la tio n o f a packaged u nifo rm m ic ro s trip line. . . .
55
5.3
A n example showing the effect o f the package on the c irc u it response. 56
5.4
Investigation o f the packaging effect on the sm all-signal analysis
by excitation o f a m o d u la te d Gaussian pulse........................................
5.5
The packaging effect on the large-signal analysis by a single-tone
excitation at 6 G H z ......................................................................................
5.6
57
The o u tp u t voltage o f the am plifier in the cases o f two different
package dimensions.......................................................................................
5.7
57
59
The spectrum o f the o u tp u t power in the case th a t a sm all-signal
Gaussian pulse m odulated at 6 GHz is imposed upon a packaged
am plifier: this exam ple demonstrates c irc u it oscillations caused by
the packaging s tru c tu re ...............................................................................
60
6.1
Flow chart o f conventional procedures for crosstalk analysis.
62
6.2
Flow chart of comprehensive sim ulation by an extended electro­
...
magnetic sim u la to r........................................................................................
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63
6.3
The configuration o f a tw o -a m p lifie r m odule........................................
64
6.4
A snapshot o f field d is trib u tio n a t the tim e step 6000.......................
65
6.5
The coupled noise voltage at (a) the near end and (b) the fa r end
by varying the in p u t pow er level at the in p u t port o f the c irc u it 1.
67
6.6
The spectrum o f the far-end noise power in Fig. 6.5 (b )....................
68
6.7
The coupled noise at the o u tp u t p o rt o f circ u it 2 by varying the
in p u t power level...........................................................................................
6.8
68
The coupled noise at the o u tp u t p o rt o f circ u it 2 by varying the
center-to-center distance D ........................................................................
69
The S-parameters o f c irc u it.1....................................................................
70
6.10 The coupling between c irc u it 1 and c irc u it 2........................................
70
6.11 The S-parameters o f c irc u it 2 showing the reduction o f crosstalk. .
71
6.12 The S-parameters o f c irc u it 1 afte r placing separators........................
72
6.9
7.1
The fields on the boundary are determ ined by the value o f in te rio r
fields, which is represented by a d ig ita l filte r........................................
7.2
The A B C configuration o f the two-m ode case by applying a bank
o f filte rs ............................................................................................................
7.3
76
77
The normalized phase constant o f a rectangular wageguide. The
system function o f a second-order IIR filte r is designed to m atch
the pre-simulated d a ta .................................................................................
7.4
79
C alculated reflection coefficients o f F D T D sim ulation and num er­
ical theory based on Eq. (7.6) using an I IR filte r o f order 2 and
3.........................................................................................................................
x
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79
7.5
A second-order I I R filte r is optim ized to m atch the norm alized
atte nuatio n constant o f a rectangular wageguide.................................
7.6
Results o f evanescent-wave absorption.
80
T h e results o f placing a
PEC on the b o u n d a ry shows the effect o f to ta l reflection for com­
parisons............................................................................................................
7.7
The calcula tion o f reflection coefficient fo r a m ic ro s trip line.
81
A
second-order I I R is used in the sim u la tio n .............................................
82
7.8
The stru ctu re o f the inductive iris...........................................................
82
7.9
The calculated reflection coefficient o f an in d u c tiv e iris in a YVR90 waveguide.
T h e A B C plane is placed a t the distance o f 0.2
A and 0.1 A fro m the iris. In the la tte r case, variours m ulti-m ode
absorption for T jEio, T E 2q, and T E 30 modes are tested. The results
are compared w ith those o f a long waveguide.......................................
xi
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83
L
4.1
is t
of
T
ables
Parameters fo r no n lin e a r elements in the c irc u it m odel
.
x ii
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A
c kn o w led g m en ts
I w o uld like to express my sincere appre ciation to m y advisor. P ro fe s s o r T a ts u o
I t o h , fo r his guidance, encouragement, and supp ort durin g m y studies at U C L A . I
am also g ra te fu l to the members o f m y D o c to ra l C om m ittee, P ro fe s s o rs B e h z a d
R a z a v i, D e e -S o n P an, and N a t h a n ie l G ro s s m a n .
T he re have been many people w ho have been instrum ental in helping me
com plete th is work. I would like to th a n k D r .
C h u n g - Y i L e e and D r . S io u
T e c k C h e w . C hung-Y i helped me a great deal b o th in m y academic and o rd i­
nary life . Siou Teck inspired the developm ent o f m y research approach thro ugh
frequent discussion. I would also like to th a n k P ro fe s s o r R u e y -B e e i W u and
J e n - T s a i K u o who provided kind advice when they were here as v is itin g schol­
ars. A lso, I w ould like to th a n k for those senior fellow students for th e ir assitance
du rin g m y "h a rd s h ip ” tim e: D r . J e n s h a n L in , D r . T ia n - W e i H u a n g , D r .
O lg a B o r ic - L u b e c k e , D r . Y a o z h o n g L iu , M r . J o h n L ia o , M r . H o w a r d
C h a n g , a n d M s . J u n e S ie y - M u n W o n g .
T he m icrowave electronics la b o ra to ry has been an incredible environm ent to
work in . The m ain reason for this was the people in the la b oratory: D r . C a r l
P o b a n z , M r . D o n g s o o K o h , M r . M i n C h e n , M r . A lf r e d P e rk o n s , M r .
K im in C h a , M s .
M a, M r.
V e sna S c h u lz , M r .
W e i Fu, M r.
A n a n ja n B a s u , M r .
B o -S h io u K e , M r .
B i l l D e a l, M r .
K u a n g - P in
J u n o K im ,
M s . S y lv ia L i n a n d M r . F e i-R a n Y a n g . Special thanks to S ylvia who spent
tim e on e d itin g m y w ritin g . F inally, I can never forget consistent supp ort and
encourgem ent in different manners fro m T I N S A , a group o f friends overseas.
x iii
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V
it a
1966
Born, Pingtung, Taiw an
1984-1988
B.S., Electronics E ngineering
N a tio n a l Chiao Tung U n iv e rs ity
Hsin Chu, Taiwan
1988-1990
M .S., Electrical Engineering
N a tio n a l Taiwan U n ive rsity
Taipei, Taiwan
1990-1992
Second Lieutenant
A rm y, Taiwan
1992-1993
Research Assistant
In s titu te of In fo rm a tio n Science, Acedemia Sinica
T aipei, Taiwan
1993-present
E lectrical Engineering D e partm ent
U niversity o f C a lifo rn ia , Los Angeles
1996
Teaching Assistant
E lectrical Engineering D epartm ent
U niversity o f C a lifo rn ia , Los Angeles
1996
IE E E M T T G raduate Fellow ship Award
P
u b l ic a t io n s
and
P
r e s e n t a t io n s
1
R.-B. W u, C.-N. K uo, and K . K . Chang, ''Inductance and resistance com pu­
tations fo r three-dim ensional m u ltic o n d u c to r interconnection." IE E E Trans.
M icrowave Theory and Tech., vol. 40, pp. 263-271. Feb.1992.
2
K . K . Chang, C .-N . Kuo, T .-L . Wu, W .-L . Chen, and R.-B. Wu, ’ E q uiva lent
c irc u it o f a thro ugh via in m u lti-la y e r environm ent.-’ Proceedings o f the 2nd
IE E E T o p ica l M eeting on E lectrical Performance of E lectronic Packaging,
pp. 59-61, 1992.
3
C.-N. K u o , B. Houshmand, and T . Ito h , ’’ S im u latio n of microwave c irc u its by
F D T D m e th o d ,” N ational Radio Science M eeting, Boulder, CO, Jan. 1995.
x iv
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4
C .-N . K u o, B. Houshmand, and T . Ito h , ” S im u latio n of microwave c irc u its
by F D T D m ethod,” the Eleventh A n n u a l Review o f Progress in A p p lie d
C o m p u ta tio n a l Electrom agnetics, vol. 2, pp. 718-723. Monterey, C A , M a rch
1995.
5
C .-N . K u o, S. T . Chew, B. Houshm and and T . Ito h . ” F D T D s im u la tio n o f
a m icrowave a m plifier,” IE E E M T T -S In te rn a tio n a l Microwave Sym posium .
vol. 2, pp. 357-260, O rlando, F L , M ay 1995.
6
C .-N . K u o, B. Houshmand, and T . Ito h , ” F D T D analysis o f active c irc u its
w ith equivalent current source approach,” IE E E AP -S Inte rna tional S ym p o ­
sium , vol. 3, pp. 1510-1513, N ew port Beach, C A , June 1995.
7
C .-N . K u o, V . A . Thomas, S. T . Chew, B. Houshmand, and T . Ito h , ” S m a ll
signal analysis o f active circuits using F D T D a lg o rith m .” IE E E M icrow ave
G u id e d Wave L e tt., vol. 5, pp. 216-218, J u ly 1995.
8
C .-N . K u o, B. Houshmand, and T . Ito h . ” Device models suitable for electro­
m agnetic c irc u its sim ulation,” Progress In Electrom agnetics Research S ym ­
posium , Seattle, W A, July 1995.
9
C .-N . K u o , B. Houshmand, and T . Ito h , ’’ S im u la tio n o f microwave c irc u its by
F D T D m ethod,” Progress In Electrom agnetics Research Symposium, Seattle.
W A . J u ly 1995.
10
C .-N . K u o, B. Houshmand, and T . Ito h . ’’ Comprehensive electrom agnetic
s im u la tio n o f active microwave circu its," A sia-Pacific Microwave Conference.
Taejon, Korea, O ct. 1995.
11
C .-N . K u o , B. Houshmand, and T . Ito h , ’’ Device models in F D T D analysis
o f m icrowave circu its,” Proceedings o f 1995 UR SI Inte rna tional S ym posium
on Signals, Systems, and Electronics, pp. 467-470, San Francisco, C A , O c t.
1995.
12
C .-N . K u o and T . Itoh, ” A filte r-lik e absorbing boundary c o n d itio n fo r
the finite-difference tim e-dom ain m ethod.” N a tio n a l Radio Science M eeting,
B o u ld e r, CO , Jan. 1996.
13
C .-N . K u o and T . Itoh, ” A n absorbing bou ndary condition for the F D T D
m eth od using d ig ita l filte r design technique.” the T w e lfth A nnual Review o f
Progress in A p p lie d C o m pu ta tional Electrom agnetics, pp. 1267-1272, M o n ­
terey, C A , M arch 1996.
xv
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14
C.-N. Kuo, R.-B. Wu, B. Houshm and, and T. Ito h , "M o d e lin g o f microwave
active devices using the F D T D analysis based on th e voltage-source ap­
proach,” IE E E M icrowave G uided Wave Lett., vol. 6, pp. 199-201, May
1996.
15
C.-N. K uo, T. Ito h , and B . Houshmand, ’’ Synthesis o f absorbing bound­
ary condition w ith d ig ita l filte r bank,” IE E E 1996 M T T -S In te rn a tio n a l M i­
crowave Symposium, vol. 2, pp. 1043-1046, San Francisco, C A . June 1996.
16
C.-N. Kuo, B. Houshmand, and T . Ito h , ’’ A p p lica tio n o f tim e-dom ain sim ­
u la tio n to microwave c irc u its ,” Progress In Electrom agnetics Research Sym ­
posium , Innsbruck, A u s tria , J u ly 1996.
17
C.-N. Kuo. B. Houshmand, and T . Ito h , ’’ A p p lication o f the F D T D m ethod
on active circuit s im u la tio n ,” IE E E AP-S International S ym posium and UR SI
Radio Science Meeting, B a lim ore, M D , July 1996.
18
C.-N. K uo. B. Houshmand, and T . Ito h . ’’ A p plication o f the F D T D m ethod
to the analysis o f housing effects in active and n o n lin e a r microwave c ir­
cuits,” Proceedings o f the 26th European Microwave Conference, pp. 537539, Prague, Czech R epublic, Sept. 1996.
19
C.-N. Kuo, B. Houshmand, and T . ito h . ’’ Analysis o f crosstalk in packaged
microwave circuits,” Proceedings o f the 5th IE E E T opical M eeting on Elec­
tric a l Performance o f E lectron ic Packaging, pp. 193-195, Napa, CA. O ct.
1996.
20
C.-N. Kuo, B. Houshmand, and T . Ito h . ’’ Full-wave s im u la tio n o f nonlinear
microwave circuits,” 4 th In te rn a tio n a l Workshop on In te rg ra te d Nonlinear
Microwave and M illim eterw ave C ircu its. Duisburg, G erm any, O ct. 1996.
21
C .-N . K uo and T. Itoh , ’’ S im u la tio n o f active microwave structures by F D T D
m ethod,” Microwave W orkshops and E xh ibition, Tokyo. Japan. Dec. 1996.
22
C .-N . K uo and T. Itoh , ’’ In co rp o ra tio n o f active devices using d ig ita l networks
in F D T D m ethod,” the T h irte e n th Annual Review o f Progress in A pplied
C o m pu ta tional Electrom agnetics, Monterey. CA. M arch 1997.
23
C .-N . Kuo. B. Houshmand, and T . Itoh . ” Full-wave analysis o f packaged
microwave circuits w ith active and nonlinear devices: an F D T D approach.”
IE E E Trans. Microwave T h e o ry and Tech.. vol. 45. pp. 819-826. May 1997.
xvi
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A
bstract
of
th e
D
is s e r t a t io n
FULL-WAVE ANALYSIS OF NONLINEAR
ACTIVE MICROWAVE CIRCUITS IN
PACKAGED ENVIRONMENTS
by
C hien-nan K uo
D o ctor of Philosophy in E le ctrica l Engineering
U niversity o f C a lifo rn ia , Los Angeles. 1997
Professor Tatsuo Ito h . C hair
Comprehensive sim ulation o f packaged nonlinear active microwave c irc u its is per­
formed by a p p ly in g the extended finite-difference tim e-dom ain ( F D T D ) m ethod.
Based on the approach of using equivalent sources, the device-wave in te ra ctio n
is characterized and incorporated in to the F D T D tim e-m arching scheme. As a
consequence, analysis of linear and no n lin e a r properties, such as harm onic gen­
eration and in te rm o d u la tio n , can be accom plished by em ploying a large-signal
device c irc u it m odel. The im plem entation is firs t validated by sm all-signal anal­
ysis o f a M E S F E T microwave a m p lifie r com paring results o f F D T D calcu la tio n
and measured data. Large-signal analysis is also performed. The s im u la tio n then
goes beyond the ca p a b ility of circ u it sim u la to rs to analyze electrom agnetic in te r­
ferences o f the packaging effect and crosstalk. Crosstalk reduction is studied by
using m etal separators to reduce substrate coupling. Besides, the technique o f
d ig ita l filte r design is employed to realize the absorbing boundary c o n d itio n in
applying the F D T D method. Consequently, the extended m ethod is applicable
x v ii
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CHAPTER 1
INTRODUCTION
1.1
P ro b lem D escription
The trend in m icrowave circuits has been towards h ig h ly integrated and m onolith ic a lly fa b ric a te d systems, such as m o n o lith ic microwave integrated c irc u its
( M M IC ’s ), co m p risin g discontinuous structures, closely spaced elements, pack­
aging structures and nonlinear active devices. In such systems, several non linear
active c irc u its are in a lig h t weight and com pact packaged module. B y ta k in g ad­
vantage o f th e features, these systems provide a ttra c tiv e applications in d ifferent
areas o f com m ercial products nowadays. Examples are wireless com m unication,
co llision-w a rn ing systems, global positioning systems, etc.
In these areas, c ir­
cu it design requires emerging techniques to predict system performance in such
com plicated systems.
T h is demand challenges microwave engineers.
W ith o u t
sufficient in fo rm a tio n , design o f microwave systems can only be accom plished by
tria l and error.
In order to achieve g e t-it-rig h t-th e -firs t-tim e fabricatio n, the design relies on
appropriate m od e lin g o f in d ivid u a l c irc u it elements and electrom agnetic effects.
M any issues should be considered in the s im u la tio n for circ u it design, in c lu d ­
ing ra d ia tio n fro m the discontinuities, m utual coup ling between c irc u it elements,
crosstalk between circu its, cavity effect of the packaging structure, in te ra c tio n
between c irc u its and the package, and action o f nonlinear active devices.
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A ll
these effects m ix and affect c irc u it performance. They m ay result in degradation
o f system performance and cause the system to fa il th e design specifications in
th e w orst cases.
C o m m ercially available design tools can be classified in to two categories,
c ir c u it sim ulators and electrom agnetic simulators.
C irc u it sim ulators, such as
H P M icrow ave Design System (M D S ) and H P-EESO F T O U C H S T O N E /L IB R A ,
a d o p t c irc u it theory approach, in w hich the S-param eter m a trix and the harm onicbala::ce m ethod are applied by d iv id in g the circu it in to standard elements and
cascading the characteristic o f each element to obtain overall system performance.
These tools have been able to design microwave c ircu its w ith great success in an
efficient manner, but, however, unable to accurately m odel the electrom agnetic
effects, w hich are ignored o r approxim ated at most. O n the other hand, electro­
m agnetic sim ulators, such as HP HFSS and SO N N ET, can fu lfill the requirement
o f in clu sio n o f the electrom agnetic effects by solving M a xw ell's equations and
ta k in g in to account the in te ra c tio n between electrom agnetic waves and circuit
elements comprehensively. T he disadvantage o f these sim u la to rs is the in a b ility
to in c o rp o ra te nonlinear active devices. To date, electrom agnetic sim ulators are
u su a lly used for passive c irc u its and circu it sim ulators are for nonlinear active
c irc u its .
Besides the m odeling o f in d iv id u a l elements, c irc u it design for these
com ple x c irc u its encounters a severe problem o f how to handle the electromag­
n e tic effects in the package m odule, in particular for no n lin e a r active circuits in
a packaging structure o pe ra ting at a higher frequency.
1.2
Literature Survey
To help solve design problems, the best policy is to develop an extensive elec­
tro m a g n e tic sim ulator capable o f device incorporation. As such, the entire sys-
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tem can be analyzed in a comprehensive sim ulation. A ccordingly, much a tte n­
tio n has been recently focused on the incorporation o f n on linear active devices
in to full-wave analysis. Some frequency-dom ain techniques u tiliz e the impedance
m a trix of the devices to achieve th is goal.
For example, the spectral-dom ain
approach (SDA), proposed in [1, 2], has been extended to analyze hybrid m i­
crowave integrated circu its w ith passive and active lum ped elements [3]; the
finite-elem ent m ethod (F E M ) has been generalized to analyze m ic ro s trip circuits
w ith a Gunn diode [4], These techniques use frequency-dom ain in fo rm a tio n , such
as S-parameters, to b u ilt up the impedance m a trix and p e rfo rm calculations in
different frequencies step by step.
Contrariwise, it is advantageous to choose tim e-dom ain techniques in those
cases th a t transient analysis is required. Am ong developed tim e-dom ain tech­
niques, the finite-difference tim e-d om a in (F D T D ) m ethod has received the most
interest for its d irect so lu tio n o f M axw ell’s equations [5].
Being generally de­
veloped, the m ethod has shown its ve rsa tility in full-wave m odeling o f complex
structures including inhomogeous m edia and com plicated shapes. A lth ough the
m ethod needs large com puter m em ory and com putation tim e, the advance o f
com puter technology has made it more and more feasible to analyze practical
problem s of large scale. The m ethod, therefore, is chosen in this dissertation for
comprehensive sim ulation o f microwave integrated circuits.
F irs t introduced by Yee in 1966 [6], the F D T D m ethod has been generally
applied to scattering problems o f calculating radiation pattern s or radar crosssection [7].
For planar m e ta lliz a tio n structures, Zhang et el.
are the first to
describe the procedure how to a p p ly this technique to analyze dispersive char­
acteristics of a m icrostrip line [8] and scattering properties o f various m icrostrip
discontinuities [9] in 1988. T he calculated results of effective d iele ctric constant
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are quite com patible w ith different frequency-dom ain approaches. Nonetheless,
this tim e-dom ain technique also benefits in the visu a liza tio n ca p a b ility o f field
propagation. In 1990, Sheen et el. presented this m e th o d as efficient and accu­
rate for analysis o f w ideband frequency responses o f microwave planar circu its
[10]. They investigated several m icrostrip circuits, in c lu d in g a low-pass filte r, a
branch line coupler and a patch antenna. Results are in good agreement w ith
measured data. Hereafter, the m ethod has been w id e ly applied to help design
microwave circuits. N o t on ly lim ite d to microwave c irc u its , it was also used to
analyze picosecond photoconductive switches [11],
Moreover, the conventional a lg o rith m was extended to be applicable for hy­
b rid systems, which are d is trib u te d electrom agnetic systems w ith lum ped c irc u it
elements. Sui et al. proposed a two-dim ensional extension and successfully im ­
plemented it in the s im u la tio n o f two-dim ensional transm ission-line circuits which
include lum ped passive elements and discrete v o lta g e /c u rre n t sources [12]. Pas­
sive components such as resistors, capacitors, and in ductors, and discrete sources
are incorporated in to the coefficients of Yee's leapfrog a lg o rith m bv specifying
adequate voltage-current characteristics o f these elements. Toland et al. general­
ized the approach to include tw o-te rm ina l nonlinear a ctive devices in d is trib u te d
circuits [13]. In th e ir paper, the lum ped equivalent c irc u it m odel o f a diode is em­
ployed and coefficients o f F D T D equations in the device region contain qua ntities
o f circu it elements. L a te r on, they also presented an a p p lic a tio n to sim ulate a
three-dim ensional microwave c irc u it which contains a G u n n diode in an m u ltip le oscillator active antenna [14]. Results can predict the correct oscillation mode.
Extended to three-dim ensional form ulation, the extensions were applied by Tsuei
et al. to the structure o f a b ip o la r transistor chip connected to a package thro ugh
bond wires [15], S im ila r to T o la n d ’s approach, c irc u it q u a ntities are embedded
in coefficients o f F D T D equations. The lum ped equivalent c irc u it model of the
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device d is trib u te s over several cells in F D T D s im u la tio n by recognizing th a t the
device m odel consists o f tw o-term inal elements. P ike t-M a y et a l also applied
a s im ila r approach to packaging interconnects in high-speed d ig ita l c irc u its w ith
active loads [16]. A more general approach, proposed by Thomas et al., substi­
tutes the device w ith equivalent current sources and consequently allows d ire ct
access to a ll S P IC E models in the F D T D ca lcu la tio n [17].
1.3
C ontributions
The w ork in th is dissertation aims at full-w ave analysis o f nonlinear active m i­
crowave c irc u its in com plex environments. Since the in te ra ctio n between electro­
magnetic waves and active devices affects system perform ance most sig nifica ntly,
it is c ritic a l to develop a general and efficient approach to perform electrom agnetic
sim ulation o f m icrowave circuits. In C hapter 2. we discuss in d etail theoretical
form ulation o f the extended F D T D a lg o rith m for th is purpose. A b rie f review o f
conventional a lg o rith m is given first. B y using the leapfrog scheme [6]. M a x w e ll’s
equations are converted to a set o f fin ite difference equations. In order to im ­
prove num erical efficiency, the equations are expressed w ith different n o tation s to
reduce the num ber o f calculations in each tim e ite ra tio n . The conventional algo­
rith m is then extended to be able to incorporate active devices of m u ltip le ports.
Following T h o m as' form ulation [17], we provide the configuration o f the F D T D
modeling for an active device. In typical cases, an active device can be lum ped
and represented by its c irc u it model. I t hence can be treated as a black box and
substituted by equivalent current or voltage sources in the device region. These
sources determ ine the value of device voltage and curre nt and subsequently lead
to update electrom agnetic fields in the device region. T h is approach is applicable
for different types o f nonlinear active devices.
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M icro strip c irc u its are very p o p u la r in in d u s try for th e ir easy fabricatio n. In
Chapter 3, th e y are chosen to dem onstrate and valida te the extended F D T D
algorithm . T he first im p o rta n t issue is the num erical e rro r from the m odeling.
Num erical tests are therefore perform ed by te stin g a sim ple configuration o f a
m icrostrip line te rm in a te d into a matched resistor to estim ate the order o f the
magnitude o f the error.
A M E S F E T microwave a m p lifie r is then analyzed to
study the perform ance o f the extended m ethod. The entire circ u it containing D C
biasing circu its and m atching networks is included in F D T D sim ulation o f sm allsignal analysis. The S-parameters are extracted by ta k in g the Fourier transform
of the observed tim e responses and compared w ith measured data. B o th examples
also dem onstrate applications to one-port devices and tw o -p o rt devices. A n o th e r
issue not addressed in previous publications is w ave-m atching in the model. In
addition to sa tisfyin g the voltage-current re latio nship at each port, the equivalent
sources must be placed to retain the same s ca tte rin g characteristics as the device.
The discussion includes the placement of equivalent sources.
Microwave c irc u its in practice are usually nonlinear. As such, there is a need
for the extended electrom agnetic sim ulato r to realize nonlinear analysis. Chap­
ter 4 discusses the procedure o f large-signal analysis using a power M E S F E T
am plifier as the p la tfo rm . I t starts w ith the d e scrip tion o f the nonlinear device
model of the M E S F E T . Linear and nonlinear properties o f the system are then
inspected. Since c irc u it sim ulators can analyze such systems, results o f F D T D
sim ulation are com pared w ith those o f HP M D S s im u la tio n to validate the im ­
plementation. T he analysis demonstrates th a t the extended F D T D m ethod is
capable o f p ra ctica l analysis and design for non linear microwave circuits.
The next consideration w ill focus on those issues o f electrom agnetic com pati­
bility/e le ctro m a g n e tic interference (E M C /E M I) problem s, which are beyond the
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ca p a b ility o f c irc u it sim ulators. The fin a l product o f a microwave c irc u it is always
placed in a packaging s tru c tu re to protect the c irc u it m echanically and electri­
cally. The package is essentially a c a v ity enclosing the c irc u it. Since c irc u it design
assumes th a t each c irc u it is e le ctrica lly isolated, the c irc u it is p o te n tia lly unsus­
ceptible to this packaging effect. Physically speaking, even though th a t unwanted
radia tion from the c irc u it can be m inim ized by proper c irc u it la yout design o f
m atching networks [18], d isco n tin u itie s in the circu it s till cause ra d ia tio n o f energy
in higher frequencies, w hich is unavoidably coupled to the resonance m ode of the
package. T his phenomenon may result in out-of-band s ta b ility problem s. Con­
ventional circu it sim ulators cannot handle three-dim ensional d iscontinu ities and
the radiation effect. In C h a p te r 5, a packaged microwave a m p lifie r is sim ulated.
The packaging effect is investigated by small-signal and large-signal analysis. A n
example is used to dem onstrate the presence of c irc u it o scilla tio n due to strong
interaction between the package and the circuit.
Another aspect o f system design is crosstalk in m u ltic h ip modules. T h is refers
to another internal electrom agnetic interference in a h ig h ly in teg ra ted system
due to near-field coupling, an unwanted coupling effect on o th e r c irc u its in close
proxim ity. T his can be a serious problem in a m ixing-signal system, where power
m ay leak from a high power c irc u it to a low noise c irc u it. Lack o f s im u la tio n tools
constraints the analysis. In C h apter 6. a packaged module w ith two microwave
am plifiers is analyzed. C rosstalk between the two am plifiers is studied by varying
different the input power level and varying the distance between the tw o circuits.
Reducing crosstalk by m eta l separators in the substrate is also discussed.
Absorbing boundary c o n d itio n (A B C ) is an im p o rta n t issue in a p p ly in g the
F D T D method to the analysis o f microwave circuits. A B C is a special a lg o rith m
employed on the boundary o f the com putation domain to reduce wave reflection
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fro m the boundary, which is nonphysical due to domain tru n ca tio n . T h is calls
for a good A B C . In Chapter 7, a technique o f using d ig ita l filte rs is developed
fo r this purpose.
Borrow ing from the filte r design technique, the approach is
im plem ented to absorb propagating and evanescent waves over a w ide frequency
range in dispersive structures. Various m icrowave structures are sim u la te d to test
its perform ance. The absorption o f m u ltip le modes is also realized by cascading
several d ig ita l filte rs w ith o u t mode e xtra ctio n .
F in a lly, conclusions are drawn in C h a p te r 8. The future developm ent is also
discussed.
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CHAPTER 2
EXTENDED FDTD METHOD
2.1
C onventional FD TD A lgorithm
The F D T D m ethod has been developed to solve for electrom agnetic problem s
since 1966 [6]. The m ethod applies the leapfrog scheme, or the central difference
scheme, to discretize tim e and space and reduces M a x w e lls differential equations
to a set o f difference equations.
The basic idea o f the leapfrog scheme is to
approxim ate derivatives by finite differences in the fo llo w in g way.
( 2. 1)
(2.2)
where the superscript and subscript denote the tim e and space index, respectively.
B y the scheme, each E-field component is located on the edge of a F D T D cell,
w hile each H -fie ld component is located on the face. The positions o f a ll six field
components fo rm an offset of a h a lf cell re la tive to the index point (i . j . k ). as
illu s tra te d in Fig. 2.1.
In the fin ite difference form, Faraday’s equation fo r free space becomes a set
o f equations.
Ac
E ? ( i , j + 1. k ) -
k)
Ay
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(2.3)
index (i,j,k)
0—
E field
H field
F igure 2.1: The p o sitio n o f each field component in a F D T D cell.
n—i . . ..
At
H r 2( L j . k ) = Hy
-------Ho
+
H rh u i-k) = H rh i,3 ,k )-—
fJ'O
+
E ?(iJ,k)-E ?(i + lJ,k)
Ax
E " [ i , j , k + 1) - E * ( t . j . k )
Az
■
(2.4)
■
(2-5)
E%( i , j , k ) - E 2 ( i , j + 1. k)
Ay
E ^ { i + l , j , k ) - E y ( i , j . k) '
Ax
w hich is applied to calculate m agnetic fields at h a lf-in te g ra l tim e steps, and.
according to Am pere’s equation, the fin ite difference equations for evaluating
e lectrica l fields at integral tim e steps are
H y +H ^ j . . k - \ ) - H y +^ { l. j. k )
Az
+
H z +Hi-3,k) -
+
Ay
- \.k)
. (2.6)
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A t \Hz + Hi - 1, j , k) - H ; +* { i . j : k)
EZ+l(iJ,k) = E%(iJ,k)+ —
eQ
Ax
+ H nx + H i , j , k ) - H : + h ( i - . j , k - 1) : (2.7)
Az
A t I Hx + ' ( L j - LA:) — Hx + 1(i,j. k)
£»
Ay
Ax
To ensure the s ta b ility o f these tim e-stepping equations, the choice o f the tim e
step must satisfy the C o ura nt co n d itio n [7],
1
Umax A t <-
(2.9)
where umax is the m axim um wave phase velocity. The space steps are usually
chosen to be sm aller th a n the wavelength as
These six equations form the core o f the F D T D m ethod. Basically, it is an ite r­
ative m ethod to sim ulate wave propagation in a flexible m anner. G iven suitable
boundary conditions and m edia parameters, it is versatile to m odel structures of
very com plicated shapes and inhomogeneous media. T he flow diagram in each
tim e advance is shown in Fig. 2.2.
2.2
Im provem ent o f N um erical Efficiency
Usually one successful s im u la tio n requires numerous tim e steps and takes long
com putation tim e to o b ta in enough tim e-dom ain in fo rm a tio n for post-processing.
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Time Step n
Time Step n + ^
( Determine H field )
Ampere’s Law
Time Step n + 1
( Determine E field )
F igure 2.2: The flow diagram o f the F D T D method in each tim e advance to
upd ate field components.
Some publications u tiliz e the technique o f signal processing to im prove efficiency,
e.g. the P rony’s m ethod in [19]. C o m p u ta tio n time, however, can be sig n ifica n tly
reduced by a simple m ethod using different notations. Each o f the six equations,
Eq. 2.3 to Eq. 2.8, contains calculations o f 4 a d d itio n /s u b tra c tio n and 3 m u ltip li­
ca tio n /d iv is io n . I f we define the voltages across a F D T D cell in each d ire ctio n as
V ei Vyi F'., and the currents as I x, I y, and
respectively, w hich equal to E xA x.
E yAy, E zA z , H XA X, H y A y, and H , A z , those equations then convert to
= i r H i . j , k ) - 2 - fr"(i.j.k ) -
+1)
+\?(i.j + l.k)-V p (i,j.k)].
(2.11)
= I y ~ h{ i , h k ) - — [K n( L ./,*:) - \ ? ( i + l . j . k )
<*y
+ \?(i.j,k + i) - V ? ( iJ . k ) ] .
(2.12)
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i? h i,i-k ) = i r h i . j , k ) - —
> - v ? u + 1.)
+ V ? ( i + h j : k) - V « ( i , j , k ) \ ,
(2.13)
1
. (2-14)
«+T,
V ^ { L j , k ) = V ^ ( i1j . k ) + A .
3„
+ l 2 + H L j A ) - I ; + 1( i J , k - 1) , (2.15)
1
v r l(ij.k) = v?(i.j,k)+n -rr;
+ ip H h j,k )-rr-(i-u ,k )
■ (2.16)
where the coefficients, a and 3. are related to space inductance and capacitance o f
the F D T D cells in different direction. T h e y are defined as. e.g. in the x -d ire c tio n .
(j,0± y A z
A xA t
e0A y A z
3x
=
AxAt
(2.17)
(2.18)
The num ber o f m u ltip lic a tio n /d iv is io n ca lcu la tio n , therefore, is reduced to be
only one in each equation.
From experience, it is found th a t the new set o f
difference equations saves about 15% to 20% o f to ta l com putation tim e.
2.3
Incorporation of M ulti-term inal A ctive D evices
In this section, we discuss the in co rp o ra tio n o f m u lti-te rm in a l active devices in a
planar m e ta lliz a tio n c irc u it. Fig. 2.3 shows a M E S F E T transistor soldered on the
substrate in a m ic ro s trip circuit and connected to the ground plane th ro u g h vias
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Transistor
Source
Gate
Drain
Source
Substrate
Grounded vias
Figure 2.3: A packaged tra n s is to r connected to the ground plane through vias at
the source p o rt in a m ic ro s trip circuit.
at the source p o rt. T h is packaged device may occupy several F D T D cells. T he
m odeling o f this n o n lin e a r active device needs to earn- out the in co rp o ra tio n o f
the device in to the F D T D tim e-m arching a lg o rith m and account for the s p a tia l
placement. A possible and com plete full-wave m ethod fo r the m odeling can a p p ly
a physical model as in [20], w hich uses a coupled system o f B oltzm ann's tra n s p o rt
equations for carrier tra n s p o rt phenomena and M a xw e ll's equations for electro­
magnetic wave propa gation. I f doing so, d iffic u lty m ay arise when the physical
model is applied to th e analysis o f a practical microwave c irc u it. U sually the size
o f meshes in the device region is required to be m uch finer than th a t required
in the passive structures o f the circuit, which results in the problem o f m em ory
lim ita tio n or n o n -u n ifo rm meshing.
A feasible m ethod is to represent the device w ith its lum ped c irc u it m odel,
while the dielectric constant in the device region is enforced to be th a t o f a ir.
Since the size o f a device is ty p ic a lly much smaller than a wavelength, this m ethod
produces reasonable a p p ro x im a tio n and s till remains a high degree o f accuracy in
full-wave analysis. Even in the case that the size o f a device is comparable to a
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wavelength, e.g.. some power devices w ith large gate w id th , the m ethod can also
be a p p lied b y c u ttin g the device region into several slices, each represented by a
lum ped c irc u it model. A lth o u g h the dimension o f a lu m pe d c irc u it is zero, the
effect fro m the spatial d is trib u tio n and the packaging s tru c tu re has already been
considered by adding parasitic elements, inductors and capacitors, in the c irc u it
m odel to account for the tim e delay as waves propagate th ro u g h the device.
Hence the key concept o f the incorporation is to connect quantities of electro­
m agnetic fields w ith quantities o f the c irc u it model. D ire ct im plem entation places
the lu m p e d c irc u it in the device region and m atch in te rn a l nodes of the lum ped
c irc u it w ith F D T D grids as used in [14, 15, 16]. Each c irc u it element, placed on
the edge o f a F D T D cell as a tw o -te rm in a l element, can be d ire c tly incorporated
in to the F D T D algorithm as the form ulation in [12, 13]. A lte rn a tiv e implemen­
ta tio n is to place equivalent sources in the device region. Reference [17] employs
equivalent cu rre n t sources. D iffe re n tly derived, a dual approach can utilize equiv­
alent voltage sources. This im p lem entation o f using equivalent sources is more
general than the direct im plem entation in the sense o f fo rm u la tio n for complex
c irc u it models and more advantageous for the m odeling o f a m u lti-p o rt device.
These equivalent sources characterize not only the voltage-current relationship
b u t also the scattering properties at each port.
2.3.1
Equivalent Current Sources
The re a liza tio n o f using equivalent current sources for the h y b rid microwave c ir­
c u it in F ig. 2.3 is shown in Fig. 2.4, where several sources are placed horizontally
on the a ir-d ie le c tric interface in order to sustain continuous curre nt flow from the
m etal s trip . One end o f each equivalent source connects to the m icrostrip line
and the oth e r to a grounded via, which provides a voltage reference as well as
15
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Vias
Microstrip
Equivalent Current Source
Gate
■y
Drain
Ground Plane
Figure 2.4: The im plem entation o f device in corp oration using equivalent current
sources.
the m od e lin g o f the vias at the source p o rt. Physically speaking, the to ta l value
o f these equivalent current sources a t each p o rt stand for the cu rre n t flow ing into
the device, o r the device current, a t each p o rt.
C onnecting field quantities and c irc u it qua ntities, the equivalent sources serve
as dependent sources, the values o f which satisfy both M a xw e ll’s equations and
the device c irc u it model. Because voltage and current relate to the in te g ra tio n of
E- and H - field, respectively, the governing equation for field u p d a tin g is derived
by ta k in g in te g ra tio n of M axw ell’s equations over those F D T D cells containin g
the equivalent sources.
The in teg ra l form o f Am pere’s equation at each p o rt
yields to
C total ~
+ I dev = ftotal*
('-T 9 )
where
Ctotal=
c,
£ c ,-,
i
=
(2.20)
(2.21)
assuming th a t there are N sources employed a t each port and C, is the space
capacitance o f each cell. Together w ith the device c irc u it model, this equation
16
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Idev,g
Idev,d
Gate
r ?\
J_Vdev,g
Itotal,g Ctotal.gJ-
Drain
Device
Circuit
V d e v ,d -l
Q\)
CtotaI,d | [total,d
Model
Figure 2.5: The N orton-equivalent c irc u it o f device-wave in te ra ctio n .
leads to an equivalent c irc u it characterizing th e device-wave interaction as shown
in F ig 2.5. The p a ir o f a current source and a shunt capacitor sym bolizes the
N orton-equivalent c irc u it o f F D T D cells as seen by the device. The tim e flow
diagram to update fields in the device region is illu s tra te d in Fig. 2.6. where Efields are determ ined by Am pere’s equation and the device c irc u it m odel. N ote
th a t the space between the gate and dra in p o rts should be large enough to avoid
unwanted free-space coupling because the tra n sfe r functions between th e two
ports are com pletely described in the device c irc u it model.
2.3.2
Equivalent V oltage Sources
Dual to the im p le m e n ta tio n of using equivalent current sources, the re a liz a tio n
o f using equivalent voltage sources is shown in Fig. 2.7. where voltage sources are
aligned to the F D T D g rid edges beneath and perpendicular to the m ic ro s trip line
w ith each ends connected to the m icro strip lin e or the grounded vias. S im ila rly ,
the value of these voltage sources represents the device voltage at each p o rt.
Assume th a t the voltage sources lie in an active sheet parallel to the x - y plane.
W ith o u t loss o f generality, consider a voltage source w ith its center denoted by
( i . j + k- k) - where tw o F D T D meshes c o n trib u te to the current flow ing th ro u g h
17
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Time Step n
Time Step n + ^
( Determine H field )
Device
Ampere’s Law
circuit
model
Time Step n + 1
( Determine E field )
Figure 2.6: The flow diagram o f the F D T D m eth od in each tim e advance to
update field components.
Equivalent Voltage Source
Microstrip
Vias
Substrate
Ground Plane
Figure 2.7: The replacement o f the active device by equivalent voltage sources
on the F D T D g rid edges.
18
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the voltage source. For the right mesh, the in teg ra l form o f Faraday’s law yields
l * A yA Z— H x { i , j + \ ' k + \ ) = ~ E y i i J +
2
’^ ) ^ + E M dik +
+ E y { i , j + \ , k + l ) A y -- E z ( i . j + 1, k + j) A - .
(2.22)
The expression can be equivalently w ritte n in term s o f circ u it quantities as
= VioopA +
V d e v
(2.23)
where the device voltage Vdev and the space inductance Ly o f F D T D cells are
defined as
V d e v
= - E y{ i J + ^ . k ) A y .
(2.24)
U =
(2.25)
Vioop,i = E : ( i . j , k + ^ ) A . + E y { i , j 4-
k + l ) A tf - E z{ i . j + 1, k + ^ ) A . (2.26)
Also,
is the equivalent voltage obtained from the loop in teg ra tion o f the E -fie ld along
the cell b ou ndary o th e r than the edge o f the voltage source, and
■fmesA.l = —H x ( i . j + —. k + - ) A X
(2.27)
is the loop c u rre n t th a t flows along the right hand side o f the active sheet through
the cell w id th A x .
A s im ila r fo rm u la tio n can be derived for the le ft mesh to obtain a c irc u it
equation for the loop current / mes/i.2 which flows along the left hand side o f the
active sheet. The device current / j e„ flow ing in to the voltage source equals the
sum o f the two loop curre nt components on b o th sides o f the active sheet. Since
the re latio nship between Vdev and /</e„ is specified by the device equivalent c irc u it
m odel, it is reasonable to construct the equivalent c irc u it as shown in Fig. 2.8.
Each mesh corresponds to a pair o f a loop voltage and a inductor.
19
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Idev
D evice
Vdev
equivalent
c irc u it model
F igure 2.8: T he equivalent c irc u it o f two F D T D meshes on the b o th sides o f a
voltage source.
In the case th a t there are N voltage sources placed across the entire w id th
o f the m ic ro s trip line, the equivalent c irc u it for each mesh can be connected
in p arallel to form a complete equivalent c irc u it for the device, as shown in
Fig. 2.9(a).
The device current
is the sum o f a ll the loop currents.
The
voltage-source approach proposed here is the dua l o f the current-source approach.
I t m ay seem th a t the equivalent c irc u it in F ig . 2.9(a) becomes more com plicated
as the num ber o f voltage sources increases. In reality, the Thevenin equivalent
theorem [21] can be applied to greatly s im p lify the com plexity o f the equivalent
c irc u it. As fa r as the device is concerned, the fin a l equivalent c irc u it can be shown
as in Fig. 2.9(b). The governing equation is derived as
(2.28)
where
(2.29)
(2.30)
and L i is the space inductance o f each mesh. The voltage source/in ductor p a ir
is the equivalent c irc u it as seen from the device.
Through this equivalent c ir-
20
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Idev
I mesh/ |L
/
I mesh!
^L.
I meshZ
Device
Vde
^ lo o p I j
Vloop31
equivalent
circuit model
(a)
Idev,g
Idev,d
Drain
Ltotal.g
Vtotal.g
Ltotal.d
Device
Circuit
Vdev.d
Vtotal.d
Model
(b)
Figure 2.9: (a) The equivalent c irc u it o f the configuration in Fig. 2.7 as seen from
the device at each port, and (b) the Thevenin-equivalent c irc u it o f device-wave
in teractio n.
21
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f
Time Step n
Device
circuit
model
Time Step n + ^
( Determine H field )
Ampere’s Law
Time Step n + 1
( Determine E field )
F igure 2.10: The tlow diagram o f the F D T D m ethod in each tim e advance to
upd ate fie ld components.
c u it. the field quantities in the F D T D a lg o rith m are combined w ith th e c irc u it
q u a n titie s in the device model to describe th e in te ra ctio n between the device and
electrom agnetic fields.
T he extended F D T D tim e-m arching scheme is illu stra te d in Fig. 2.10. which
can be described as follows:
S te p 0: T he electric fields E n~ l , m agnetic fields H n~ T device voltage V^ev- .
and device current I j ev2 are given, where the superscript denotes th e tim e
step.
Step 1: T he update o f electric fields E n is form ed from E n~ l and H Tl~ i by
e m plo ying A m pere’s law. Note th a t the electric fields on the active sheet
have yet to be determined since A m pere's law is not applicable in the device
region.
22
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S te p 2 : The voltage source V£tal is calculated from nearby electric fields by
Eq. (2.29). The F D T D meshes as seen from the device satisfy Eq. (2.28).
C o m bining Eq. (2.28) and the device circu it model, the values o f the device
voltage and current at the next tim e step. V ^ 1 and I ^ v 2. can be solved by
using typical c irc u it approaches w ith the in itia l values o f
and
respectively. The electric fields E n on the active sheet are then determined
by the device voltage V£v w hich is approxim ated by the average o f V '^ 1
and
S tep 3: The magnetic fields H n+5 are updated from H n~? and E n by em ploying
I t is w o rth noting th a t Step 1 and 3 in the updating a lg o rith m for the electric
and m agnetic fields are exactly the same as those in the conventional F D T D
a lg o rith m . Regardless of how com plicated it may be. the device c irc u it model
can be viewed as a black box having no influence on the F D T D a lg o rith m in the
field calcula tion. Nonetheless, the device c irc u it model affects the solution of the
device voltage and current in Step 2, w hich is accomplished solely in the circu it
level. In th is way. the approach a d m its d ire ct access to all available c irc u it models
for th e lum ped devices in the F D T D sim ulation. The device models in circu it
sim ulato rs like SPICE can be used d ire c tly in the F D T D sim u la tio n w ith o u t the
need to repeat the model development.
2.4
Differential E quation Solver
In each tim e advance, the device voltage is evaluated from the state equation of
the c irc u it in Fig. 2.5 or Fig. 2.9 and subsequently used to update the electromag­
netic fields in the equivalent-source region. The state equation can be generally
23
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expressed as a nonlinear d iffe re n tia l equation by
A(X) ~
= B (X ) • X + F(X),
(2.31)
where the vector X denotes the sta te variables, the matrices A and B are derived
from the c irc u it elements, and th e forcing term F comes from the source, I t otai
or Vt0tai■ P erform ing the backward difference scheme yields to a fin ite difference
equation by
G ( X n+1) = —
• ( X „ +1 - X „ ) - B ( X n+1) - X n+[ - F ( X n+i) = 0,
(2.32)
where the subscript denotes the tim e step. Afterw ards, ite ra tive searching for
X n+i is perform ed by the N ew ton-R aphson m ethod [22], a m u ltid im e n sio n a l root
fin d in g m ethod. Given X „ as the in it ia l value. X n+i is calculated ite ra tiv e ly u n til
it converges by
X £ ‘, = X j+1 - (J(X„+1)]-'
■G(X;,+I).
(2.33)
where J is the Jacobian m a trix and its elements are defined as
J„ S g .
(2.34)
The c rite rio n o f numerical s ta b ility is n o t on ly to satisfy the Courant c o n d itio n in
the F D T D a lg o rith m b u t also to choose A t such th a t J is not singular. T y p ic a lly
the A t chosen fo r the form er c rite rio n is in the order o f pico-seconds for m icrowave
circuits, which is much sm aller than the A t required for the la tte r c o n d itio n w hich
is in the order o f nano-seconds. T hu s there is no additional num erical b u rde n o f
needing to choose a smaller A t.
24
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CHAPTER 3
SMALL-SIGNAL ANALYSIS OF
AMPLIFIERS
3.1
In trod u ction
The microwave a m p lifie r is a common com ponent in microwave circu its and
w idely used for signal am plification.
In the a m p lifie r, a M E S F E T tra n s is to r
is usually used as the active device in the signal p a th to a m p lify the signal power.
According to the device S-parameters, m atching c irc u its are designed at the in ­
p u t/o u tp u t p orts fo r m axim um power transfer o r low noise figure [23]. In prac­
tica l design, c irc u it sim ulators are used to o p tim iz e the m atching circu its in the
environm ent w ith o u t substantial electrom agnetic influence and able to p re d ict
system perform ance successfully.
A m plifiers, therefore, are chosen to dem on­
strate the equivalent-source approach as discussed in C hapter 2 and to study its
performance and characteristics.
In th is C hapter, a microwave am plifier is designed and fabricated. The ex­
tended F D T D m eth od is applied to perform sm all-signal analysis of this a m p lifie r.
Results o f F D T D sim ulations are compared w ith measured data. In Section 3.2,
a m icro strip line te rm in a te d into a matched resistor is first sim ulated to s tu d y
the order o f num erical errors from the model o f device incorporation. It is also an
example in tro d u c in g the application to one-port devices. Small-signal analysis
25
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Microstrip line
Substrate
50Q
Ground
Figure 3.1: The side view o f the configuration o f a 50-f i m icrostrip line te rm in a te d
w ith a 50-D resistor.
o f the m icrowave a m p lifie r is performed in Section 3.3. Essentially this a m p lifie r
served as an exam ple o f microwave circuits w ith tw o -p o rt devices. C alculated re­
sults are verified by comparison to measured d a ta . T he next consideration is the
scattering p ro p e rty in the modeling. In a d d itio n to the voltage-current re la tio n ­
ship, those equivalent sources must retain the same scatte ring characteristics as
the device. The connection between the sources and the m icrostrip line results in
discontinuity. T h e effect o f the discontinu ity fro m cu rre n t or voltage m ism atch ing
w ill be discussed in Section 3.4.
3.2
M icrostrip Line Term inated into a M atched R esistor
N um erical tests are perform ed to estim ate the m odeling error of device in co rp o ­
ration. A u n ifo rm m icro strip line designed to have a characteristic impedance o f
50 H at 6 GHz is term in ate d into a 50-fi resistor.
Fig. 3.1 illustrates the side
view o f the c o n fig u ra tio n in the F D T D analysis. T he sim ulation is perform ed
w ith u n ifo rm grids o f space steps A x = 13 m il, A y = 7.5 m il, and A , = 10 m il.
The entire c o m p u ta tio n dom ain is divided in to a g rid o f 38 (w id th) x 25 (heig ht)
x 230 (length) cells. The m icrostrip conductor is d iv id e d in to 7 cells in the w id th
direction. The substrate is divided in to 4 cells in the height direction. H igdon's
26
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co
TJ
co
to
£
c
3
0)
QC
Voltage-source approach
Current-source approach
-10
-20
-30
-40
0
5
10
Freq (GHz)
15
20
Figure 3.2: The re tu rn loss o f the m ic ro s trip c irc u it in Fig. 3.1.
second-order absorbing bou ndary condition is used as in [43]. Eight equivalent
sources are placed beneath the m icrostrip line and connected to the ground plane
through vias. These sources m aintain the voltage-current relationship o f 50 Q. A
m odulated Gaussian pulse is excited to observe the reflection from these sources.
The calculated re tu rn loss is shown in Fig. 3.2, where results sim ulated w ith
the current-source and voltage-source approach are plo tte d .
Both results are
w ith in the same order o f m agnitude in the tested frequency range. Ideally, the
reflection should be as sm all as possible. In the m odel, the device voltage is uni­
form over the w id th o f the m icrostrip line, which does not exactly satisfy the edge
effect o f the m ic ro s trip conductor and causes discontinuities o f the electrom ag­
netic wave. Those vias used to connect the equivalent sources to the ground plane
also unavoidably in tro d u ce inductive effects. Furtherm ore, the m icrostrip line,
a dispersive waveguiding structure, is s tric tly 50 f2 at 6 GHz only, ra th e r than
over the entire frequency band o f interest. These factors result in an impedance
27
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
m ism atch and, subsequently, a sm all reflection from the te rm in a tio n resistor.
3.3
Sim ulation of a M icrowave Amplifier
In this section, sm all-signal analysis o f a microwave a m plifier is carried ou t. This
am p lifie r is designed to have 9 dB gain a t 6 GHz. Fig. 3.3 illu s tra te s the circu it
and its layout. D C biasing circu its and m atching circuits are in clu d e d in the sim­
u la tio n . T h e entire system contains three types o f structures: d is trib u te d passive
structures, lum ped passive devices, and an active device (G aA s M E S F E T , NE
72084). T h e d is trib u te d passive structures are sim ulated using the conventional
F D T D a lg o rith m . T he radial stubs in the D C biasing circuits is m odeled by the
staircase a p p ro xim a tio n . The lum ped passive devices, resistors and capacitors,
in biasing c irc u its and m atching c irc u its are modeled as d is trib u te d elements and
incorporated in to coefficients o f the F D T D algorithm as in [12].
The m odeling o f the M E S F E T, a tw o -p o rt active device, applies the equivalent
source approach. T h is device is s u b stitu te d w ith equivalent c u rre n t or voltage
sources, w hich represents the current-voltage relationship at the device term inal
and characterizes the in put impedances as well as the transfer fun ctio n s o f the
active device. The lumped c irc u it m odel o f the packaged M E S F E T is depicted in
Fig. 3.4, w hich is a sm all-signal m odel w ith linear elements. The packaging effect
is modeled w ith parasitic elements, in ductors and capacitors.
Those element
values are o p tim iz e d to m atch the measured S-parameters o f the active device
biased at VD = 3 V and I DS = 30 m A .
F D T D sim u la tio n s are perform ed w ith u nifo rm grids. The d ie le c tric constant
for ca lc u la tin g E-fields on the air-d ie le ctric interface is assigned as the average of
28
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(a)
DC block
capacitor
-li-
DC block
capacitor
(b)
Figure 3.3: (a) The picture, and (b) the layout o f the microwave am plifier,
signed at 6 GHz, for sm all-signal analysis.
29
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Lgg
q
L§
o—
Rg
Rd
Cgd
Ld
Ldd
H ftT l-r-fT F P —o p
>a a j - — if—
Cgs:
gm(?)
Cgsp i
^
Ri
<
=pCds
Rds
- p Cdsp
Rs
Ls
gm=48.86 ms
Unit: £2, pF, nH
Rg=1.71
Cgs=0.553
Lg=0.548
Rd=1.27
Cgd=0.06
Ld=0.589
Rs=0.5827
Cds=0.073
Ls=0.20
Ri=0.856
Cgsp=0.1668
Lgg=0.01
Rds=282.56
Cdsp=0.226
Ldd=0.47327
Figure 3.4: The device c irc u it model o f the packaged M E S F E T .
30
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the constant in the a ir and the substrate,
t'H =
(3.1)
On the b o tto m faces in the com p u ta tio n dom ain is the ground plane, and on the
others is th e second-order H igdon’s absorbing boundary co nd ition as described
in [43].
Power pads connect to the ground plane through vias to m odel RF
short c irc u its .
used are A * =
In the case o f using equivalent current sources, the space steps
13 m il,
is A t = 0.594 ps.
= 15 m il, and A ; = 10 m il. and th e tim e step
The entire co m p u ta tio n dom ain is divided in to a g rid o f
9 1 (w id th ) x 12(height) x 195(length) in x, y, and
2
directions, respectively. Those
parameters for using voltage sources are the same except th a t A y = 7.5 m il
and the d o m a in is 90 (w idth) x 25 (height) x 230 (length).
T h e packaged
M E S F E T occupies 8 F D T D cells in the lo n g itu d in a l direction and has contacts
which extend over the entire w id th o f the m icrostrip lines.
For th is tw o -p o rt
device, tw o sets o f equivalent sources are placed at those cells connected to the
m icro strip lin e at the gate and drain p o rt. Those sources are placed on the airdielectric interface horizontally for current sources, and are placed beneath the
m icro strip lin e v e rtic a lly for voltage sources. The numbers o f sources a t the gate
and dra in p o rts are 8 and 4. respectively.
The voltages from the gate to the
ground plane and from the drain to the ground plane are used.
Source e x c ita tio n employs the scheme o f using soft sources [24] ra th e r than
hard sources, w hich enforce values to E-fields on the source plane.
The soft
source is advantageous in th a t it is transparent to reflected waves so th a t the
source plane can be placed very close to discontinuities w ith o u t causing a d d itio n a l
p e rtu rb a tio n . In num erical fo rm ulation, it is an added current source in Am pere's
equation.
0 T7
V x H = e—
+ J soft.
31
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(3.2)
The soft sources are u n ifo rm ly d is trib u te d under the m icrostrip and specified in
the fo rm o f m o dulated Gaussian pulses.
2'
(3.3)
where ui is the center frequency, chosen as 6 G H z here.
There are 6000 tim e steps in one s im u la tio n . Fig. 3.5 displays snapshots o f
field d is trib u tio n d u rin g sim ulation at d ifferent tim e steps. These pictures show
how waves propagates in the c irc u it region.
The incident wave arrives a t the
in p u t p o rt, enters the c irc u it region, and leaves the c irc u it am plified. M eanw hile
discontinuities result in reflected waves.
Consecutive snapshots give physical
insight and clear visualization of the in te ra ctio n between electrom agnetic waves
and c irc u it elements in this microwave c irc u it.
For ca lcu la tin g the tw o-port S-parameters. the reference planes a t the in ­
p u t/o u tp u t p o rts are 300 m il from the D C -b lo ck capacitors.
observed tim e response at the in p u t/o u tp u t ports.
Fig. 3.6 is the
C alculation o f S-param eters
follows the procedure described in [9]. The tra n s m itte d field is obtained fro m the
observed tim e responses at the o u tp u t p o rt, w hile the reflected field is obta in e d
by su b tra c tin g the incident field, available from a pre-sim ulation o f a u n ifo rm
m icro strip line, from the observed field at the in p u t port.
B y ta k in g Fourier
transform w ith zero-padding on the reflected and tran sm itted fields, the ra tio o f
these fields to the incident field leads to S-param eters over a wideband frequency
range,
(3.4)
(3.5)
where subscripts r. t and i denote the reflected, tran sm itted and in cid e n t fields,
respectively. T he gain at 6 GHz, using the equivalent current sources, is 9.3 dB
32
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(b)
F ig u re 3.5: Snapshots o f field d is trib u tio n at different tim e steps, (a) 320. (b)
640, (c) 960, and (d) 1280. ( C ontinued)
33
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(c)
Figure 3.5: ( C ontinued)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4
Input port
Output port
2
o
CD
CT>
ffl
3>
0
■a
CD
£
03
to
-Q
O
■2
i
-4
2000
4000
6000
Time steps
F igure 3.6: The observed tim e responses at the in p u t/o u tp u t ports o f the m i­
crowave am plifier.
35
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
and, The m atch p o in t is 5.7 GHz w ith the cu rre n t sources approach. 5.75 G Hz
w ith the voltage source approach, and 5.6 G H z w ith measurement.
M easured
data and th e results o f using current-source approach are p lo tte d in F ig . 3.7
for com parison.
In Fig. 3.7(a), measured d a ta less than -2 dB are tru n c a te d
because o f the choice o f the scale in experim ent.
The results o f the voltage-
source approach and the current-source approach are very sim ilar, since b o th
approaches are dua l o f each other. The spike in the (Soil curve at ab o u t 1 G Hz
is related to low-frequency oscillation. I t has been found th a t the m o d e lin g o f
the D C power supp ly strongly affects the m agnitu de of the spike. In any case.
F D T D sim ulations can predict those o u t-of-ban d dips near 1 GHz and 11 G Hz
in the |5 n | curve. C alculated results agree well w ith measured data.
3.4
Effect o f M ism atching
Since the equivalent sources represent the signal reflection and the transm ission
characteristics o f the active device, they should be d istrib u te d over a ll the cells
across the m ic ro s trip line to avoid a d d itio n a l m ism atching from the connection
between the m ic ro s trip line and the equivalent sources. In this Section, we discuss
the effect o f d iffe re n t numbers o f equivalent sources used in the m odeling w h ile
the device voltage and current rem ain the same.
In Section 3.3. the numbers o f equivalent sources used at the gate and d ra in
ports are fixed and d istrib u te d over the entire w id th o f the m icro strip lin e . Here,
several sim ulations are performed by choosing different numbers o f equivalent
sources, and the calculated results are p lo tte d in Fig. 3.8.
W h ile m a in ta in in g
the same voltage-current relationships, three d iffe re n t sets o f equivalent voltage
sources are placed at the gate and dra in ports. T he first case is to place sources
over the whole w id th o f m icrostrip line, 8 at the gate port and 4 at the d ra in
36
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Voltage-source
Current-source
Measurement
10
0
-1 0
0
6
3
9
12
Freq (GHz)
(a)
Voltage-source
Current-source
Measurement
CQ
T3
-20
0
6
3
9
12
Freq (G H z)
(b)
Figure 3.7:
The calculated results of two different approaches and measured
S-parameters, (a) I-S2 1 1, and (b) |S n |.
37
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
port. The num ber o f equivalent sources is then reduced fro m 8 to 4 to 2 at the
gate p o rt, and 4 to 2 to 1 a t the d ra in p o rt. Calculations show th a t the m atching
point o f the |SU | curve shifts down fro m 5.75 GHz to 5.35 G H z to 5.03 G H z as
the num ber o f sources is reduced. The sh ift of the |S2i| curve shows a sim ilar
trend. In a d d itio n , the v a ria tio n o f different numbers has effect on the response
o f the S-param eters m o stly in the center frequency range.
38
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
25
- Gate:8, drain:4
- Gate:4, drain:2
- Gate:2, drain: 1■o Measurement
m
■o
co
-15
0
6
3
9
12
Freq (GHz)
(a)
o
Gate:8, drain:4 Gate:4, drain:2_
Gate:2, drain:1
Measurement
m
-o
co
-15
-25
0
3
6
Freq (GHz)
9
12
(b)
Figure 3.8: The calculated and measured S-parameters. (a) |S2i|. and (b) |S n |.
The numbers o f voltage sources used to represent the gain and d ra in p orts o f the
device are chosen as parameters.
39
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER 4
LARGE-SIGNAL ANALYSIS OF
AMPLIFIERS
4.1
In trod u ction
In reality, an active device is ty p ic a lly nonlinear. The microwave c irc u it co n ta in ­
ing the device provides necessary biasing cond ition s such th a t the device works
at a specified o pe ra ting point. The in itia l design uses the device S-param eters at
this o p e ra tin g p o in t. Sometimes c irc u it design needs to change different o p e ra tin g
points for th e best tu n in g condition o r the m a xim u m tunable range [25]. In order
to analyze th e microwave circu it generally, it is necessary to perform large-signal
analysis by in c o rp o ra tin g the nonlinear device c irc u it model in to electrom agnetic
sim ulations.
In this C hapter, a power M E S F E T a m p lifie r is analyzed. Results o f F D T D
sim ulations are compared w ith those o f c irc u it sim ulators.
The device c irc u it
model is described in Section 4.2, where th e state equation o f the device is for­
m ulated. N o n lin e a r analysis is perform ed in Section 4.3. Those quantities related
the design process, such as harmonics generation and interm odulation, are exam ­
ined. S m all-signal analysis is also realized using the nonlinear device m odel by
pum ping a sm all in p u t power.
40
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Ltotal.g !
Lg
Rg
(j
Cgd
j-j ,
—nnrwvs—
11
„ _L+ Ir
!G
Rd
v \a
C gs_vg
Cds
Id s ®
Ld
—
; Ltotal.d
-p t j t s
Dj
;
Rds>
Ri
S
Q v to ta l.g
Vtotal.d^^
Rs
Ls
<
Rg = 0.5 Q
Rd = 0.5 £2
Lg = 0.05 nH
Rs = 0.7 Q
Ld = 0.05 nH
R i= I.O Q
Cdg = 0.2 pF
Cds = 0.6 pF
Ls = 0 . 1 nH
Figure 4.1:
Inside the dashed box is the large-signal device circu it m odel o f
a M E S F E T used in this paper.
The gate-source capacitor Cgs and the d ra in
current source 7^, are nonlinear.
4.2
D evice C ircuit M odel
The c irc u it m odel o f the M E S F E T for large-signal sim u la tio n is shown inside
the dashed box in Fig. 4.1. which also includes the Thevenin-equivalent c irc u it
o f the F D T D cell. The in terna l nodes G ’D ’S’ represent the in trinsic part o f th e
M E S F E T . D iffe re n t from the circu it model in Fig. 3.4, this c irc u it model contains
two nonlinear elements, the gate-source capacitor C gs and the drain current I ds.
Governed by the P N -ju n ctio n capacitance m odel, the gate-source capacitor is
expressed as
(4.1)
41
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0
2
4
6
8
10
VDS( Volts)
Figure 4.2: D C characteristics of the M E S F E T .
Cgso
*Pbi
-4o
-4,
■4-2
-4.i
a
3pF
0.7V
0.5304
0.2595
-0.0542
-0.0305
1.0
Table 4.1: Parameters fo r nonlinear elements in the c irc u it m odel
T he d ra in current I ds, describing D C characteristics, relates to vg and v d as
I d s { v g: ud )
= (*4o
4 - A \ V g 'S '
+ .4.9Uq,s, + A^Vq,s, ) tanh(ci:L'(/ ) .
(4.2)
Those parameters are listed in T able 4.1. The DC characteristic is p lo tte d in
Fig. 4.2.
A fte r choosing state variables as [vg, vg’ d ' • <-’</•
it is stra ig h tfo rw a rd
to derive the state equation fro m the nodal equation o f the c irc u it in Fig. 4.1.
42
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The resultant matrices in the state equation are found as
A =
C g s{vg)
0
0
0
0
Cgs ( ^g)
Cgd
0
0
0
0
C g d
— Cds
0
0
<?,[(%■ - DL'g - % L S|
0
0
0
0
0
0
G
d
[ %
L
' g
-
%
L
G
S \
g
[ %
C d [{ % -
L
' d
-
1) L
-
%
d
-
L
S ]
% -i
(4 .3 )
Cgs
Cgs
0
0
0
0
0
1
0
0
0
0
0
-1
- C
B =
gs
0
G a{ ! - %
■ )
Si
1
0
r
Qs.
0
1
^ 9 G
- G d%-
0
r
(4.4)
0
0
F =
(4.5)
I d s ( Vg ■ V d )
G g( 1 ~
^ y )^ to ta l,g
— C d ~ G ^ to ta l,g + G
_
(/ ( 1
(7 S i
'“ S G I ' to ta l .(/
— ^ f)\to ta l.d
where some notations are defined as
Lg
and
Lg
+ Ltgtal^g.
Ld
=
L d 4 ' L to ta l,d -
G
=
G
—
1/
G
g
+
g
R
g :
G
d
-F
G
d
G
s.
= 1 /^ ;
G
s
=
1/
R
s .
Note th a t B is independent o f the state variables. The iterative fo rm u la for X n+i
is
X fi1, = X i+1 -
+ A) - B -
ax„.,
• f(X J
n tl).
43
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4.6)
D e vice re g io n
Drain
Ro
Pin
Vgg
Figure 4.3: T he stru c tu re of a microwave a m p lifie r fo r large-signal analysis.
The device voltage is evaluated by
(4.7)
where i means the gate o r the dra in port.
4.3
F D T D Sim ulation
The extended F D T D m ethod is applied to an electrom agnetic sim u la tio n o f the
power a m p lifie r as shown in Fig. 4.3. T his a m p lifie r excludes biasing c irc u its , so
D C biasing is established by d ire ctly a pp lying D C voltages at the in p u t/o u tp u t
ports. The size o f the M E S F E T, which resides in the region o f 80 m ils in the
lo n g itu d in a l d ire c tio n , is much sm aller than the guided wavelength at 6 G H z.
In F D T D sim ulations, the m icrostrip lines at the in p u t/o u tp u t ports are con­
nected to voltage sources, Vgg and Vm , each w ith a source impedance o f 5 0 -fl.
D ifferent from using soft sources as in Section 3.3, source excitatio n is established
using these voltage sources. The d e ta il of the config uration is shown in Fig. 4.4.
The height o f the substrate is divided into 4 F D T D cells and there are 10 cells
in the w id th o f the m ic ro s trip line. A lto gether there are 4 x 10 voltage sources
44
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Microstrip
Ground
Figure 4.4: The configuration o f source e x cita tio n .
under the m icro strip , each w ith a source impedance. T h e cascaded to ta l voltage
and impedance o f these sources, connected in parallel and in series, equal to \'giJ
or V^d and 5 0 -fi. The m odeling o f each voltage source is as form ulated in [12].
The sim ulation starts w ith D C biasing. DC voltages are set at Vgg and Vm - An
A C power is then pum ped in to the input p o rt afte r the c irc u it has reached the
steady state. Those q u a n titie s for circuit design exist in th e steady-state response.
In perform ing large-signal analysis, a large num ber o f tim e steps is required to
ensure th a t the tim e sequence long enough for post-processing procedure of tak­
ing Fourier transform o f the steady-state response. The s im u la tio n adopts 15000
tim e steps.
Fig. 4.5 displays a snapshot o f field d is trib u tio n . T h e p lo t characterizes the
biasing co nd ition o f th e gate p o rt, which is biased at a negative voltage of I'as
= -0.81 V', and the d ra in p o rt, biased at a positive voltage o f I'o s = 6.4 V. The
voltages at the in p u t/o u tp u t ports are recorded. The observed tim e responses
are shown in Fig. 4.6. T he c irc u it moves from the tran sien t state in to the steady
state around the tim e step 4000. As can be seen, a spike appears at the output
p o rt at the beginning o f D C source excitation due to wave reflection from the
45
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 4.5: A snapshot o f field d is trib u tio n d u rin g sim u la tio n .
10 |
\
8
■
—
Transient state
++
6
M
o
>
Drain voltage
4
2
Gate voltage
0
A/v w
2
0
v
W
5000
w
w
J
a
/w
v
:
10000
Time steps
Figure 4.6: The observed tim e response at the in p u t/o u tp u t ports.
46
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
m atching stu b . I f the voltage spike is too large, there is the p o s s ib ility o f causing
break down damage to the active device.
A lth o u g h the nonlinear large-signal m odel is employed, sm all-signal analysis
is s till available. A Gaussian pulse m o dulated at 6 GHz is used w ith a m p litu d e
sm all enough to allow the c irc u it to operate in the linear region. The incident
wave is the measured voltage in the system o f a sem i-infinite m ic ro s trip line.
A C responses are obtained by su b tra c tin g the tim e responses to those o f pre­
sim ulated D C -o n lv excitation. The S-parameters are then calculated from the
reflected and tra n sm itte d waves s im ila r the procedures in C h a p te r 3. A c irc u it
sim ulato r, H P M D S, is also applied to sim ulate the circuit. P a rtic u la rly , the vias
at the source p o rt are modeled by a series in d u c to r to the gound o f 0.05 nH , which
is a typ ica l value o f effective inductance. A ccording to the device m odel used in
F D T D sim u la tio n , the C urtice cubic m odel and the ju n c tio n cu rre n t m odel are
chosen from the HP MDS lib ra ry, and suitable parameters are set. B o th results
o f F D T D and HP M DS sim ulation are p lo tte d in Fig. 4.7 for com parisons. The
m atching d ip o f F D T D sim ulation is at 5.58 GHz. deviating fro m th a t o f HP
M DS s im u la tio n by 7% in th is case. T h is d e via tio n may come fro m the m odeling
o f the vias a t the source port. The frequency o f the m atching d ip is sensitive to
the effective inductance of the vias, w hich cause series feedback effect. I t is found
th a t, in HP M D S sim ulation, the frequency o f the m atching d ip decreases as the
inductance decreases.
The nonlinear property o f the a m p lifie r is inspected by eva lu a tin g the o u t­
p u t power, w hich is the dissipated power calculated from the voltage across the
loading resistor R i by the d e fin itio n o f
(4.s)
R-l
where frequency-dom ain in fo rm a tio n is obtained by taking Fourier tra n sfo rm of
47
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
20
FDTD
HP MDS
CD
TJ
-10
-20
-
2
6
4
8
10
Freq (GHz)
Figure 4.7:
S m all-signal analysis o f the microwave am plifier a p p ly in g the
large-signal c irc u it model.
48
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
50
P„ = -4.04 dBm
P„ = 5.95 dBm
P„,= 13.9 dBm
E
CD
n
CL
-50
-100 L
X
X
Freq (GHz)
Figure 4.8: The spectrum o f the o u tp u t power using single-tone e xcitatio n o f
different power levels at 6 GHz.
the steady-state response. Single-tone e x c ita tio n produces o u tp u t power at har­
monics due to the nonlinearity. Fig. 4.8 illu s tra te s the spectrum o f the o u tp u t
power w ith different levels of the in p u t power at 6 G Hz. The power o f harm onics
increases ra p id ly as the input power increases. N ote th a t the o u tp u t power ap­
pears at harm onic frequencies only. The power between harmonics is a ctually the
num erical noise from the Fourier transform . A p p ly in g larger tim e sequences can
lower down the num erical noise. Fig. 4.9 shows the curves o f the o u tp u t power
to the in p u t power for different harmonics. T he power at the 1-dB compression
p o in t is 25.1 dB m .
In te rm o d u la tio n is determined by tw o-tone e xc ita tio n at 3 GHz and 6 G H z
w ith the same in p u t power level. The spectrum o f the o u tp u t power for different
in p u t power levels is plotted in Fig. 4.10.
The o u tp u t power appears o n ly a t
49
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
50
—x FDTD
- HP MDS
fundamental
E
CD
■o
3
o
2nd
CL
-50
-100
-10
0
10
20
P,„ (dBm)
Figure 4.9: The o u tp u t pow er o f harmonics using single-tone excitation.
the frequency of m ixin g frequencies, or interm odulation p ro d u cts, which arise as
lin e a r com binations o f 3 G H z and 6 GHz. For low in p u t power, the numerical
noise in this analysis m ight be too large to obtain accurate o u tp u t power. The
o u tp u t power o f different in te rm o d u la tio n products is show n in Fig. 4.11.
These
results o f nonlinear analysis b y F D T D sim ulation are in good agreement w ith
those obtained by HP M DS s im u la tio n . The analysis o f these system responses
verifies the ca pa bility o f the extended F D T D m ethod in de a lin g w ith nonlinear
active microwave circuits.
50
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
50
P„ = -4.03 dBm
P„ = 2.06 dBm
P = 1 6.0 8 dBm
0
-50
-100
3
6
9
15
12
18
Freq (GHz)
Figure 4.10: T he spectrum o f the o u tp u t power using two tones o f the same
power level P in a t 3 GHz and 6 GHz.
40
3 GHz .
6 GHz
9 GH? - '
-40
12 GHz
-80
■* FDTD
HP MDS
3
o
a.
-120
-10
0
10
20
P,„ (dBm)
F igure 4.11: The output power o f in te rm o d u la tio n products.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER 5
INVESTIGATION OF PACKAGING EFFECT
5.1
In trod u ction
W hen the c irc u it is enclosed in a packaging s tru c tu re , an interesting question
arises, i.e., how the packaging structure affects c irc u it performance. C h aracteri­
zation o f the packaging effect is beyond the c a p a b ility o f c irc u it sim ulators but
can be accom plished by the extended F D T D m eth od by including the c irc u it as
well as the packaging structure in the analysis as a whole.
Physically, the packaging structure forms a p a rtia lly -d ie le c tric -fille d cavity.
Excited by the c irc u it, the cavity stores energy due to the n atural resonance and
the stored energy is in e v ita b ly coupled back to the c irc u it. For an am plifier, this
feedback makes the s ta b ility circles d rift and m ay result in oscillation. In order to
avoid oscillation, dimensions of the packaging s tru c tu re are usually chosen such
th a t the resonant frequency is raised far above the frequency range o f interest.
The d iffic u lty w ith the characterization o f packaging structures results from
the fact th a t the s tru ctu re is semi-open ra th e r th a n com pletely shielded and
there are non-planar discontinuities.
Thanks to the development o f full-wave
techniques, the analysis o f a three-dimensional com plex package is available by
theoretical analysis. For example, the F E M m eth od was u tilize d to analyze an
herm etic package [26]; the F D T D m ethod was used to analyze via-hole grounds
enclosed in a package [27]. The configuration o f the packaging box can be ad-
52
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ju s te d and designed so th a t the resonance modes are above the frequency o f
operation- Moreover, design process can combine electrom agnetic sim ulators and
c irc u it sim ulators. The equivalent c irc u it o f the package is obtained by an elec­
tro m a g n e tic sim ulator and then cascaded w ith the microwave circu it in a c irc u it
s im u la to r fo r the purpose o f c irc u it o p tim iza tion. References [28. 29] dem onstrate
the procedure of param eter e x tra c tio n for the equivalent c irc u it model o f a pack­
age.
In Reference [30], a broadband am plifier IC m odule is designed by using
the m u lti-p o rt network m odel o f the package to sim ulate m odule performance.
Nevertheless, the extended F D T D m ethod is able to fa c ilita te comprehensive
analysis and design o f the entire system. In Section 5.2. a packaged microwave
a m p lifie r is analyzed to s tu d y the packaging effect on the c irc u it performance.
A n in te re stin g example is given in Section 5.3 to dem onstrate oscillation due to
the electrom agnetic interference from the package.
5.2
Sim ulation o f Packaging Effect
The configuration o f the system under consideration is shown in Fig. 5.1.
As
one can see. a microwave a m p lifie r is enclosed in a package, which is modeled as
a P E C box w ith two holes a t the in p u t/o u tp u t ports for feeding power. Those
connectors o r transition structures are not included in the sim ulation.
T h is a m plifier is the same as th a t used in C hapter 4 and is also connected
to voltage sources at the in p u t/o u tp u t ports pro vid in g D C biasing conditions
and in p u t power. In order to avoid oscillation, the dimensions o f the packaging
box are chosen to be 640 m il x 186 m il x 690 m il. To determ ine the resonance
frequencies, the structure o f a u n ifo rm m icrostrip th ro u g h line in the package
is pre-sim ulated by F D T D . T he p lo t o f the insertion loss is shown in Fig. 5.2.
illu s tra tin g th a t the first resonance frequency is at 11.79 GHz. Those ripples near
53
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Packaging box
Device region
TlVdd
90
NX
n
RS
Input port
^-8CT
N180
output port
160
50
Pin
Unit: mil
Figure 5.1: The stru c tu re and dimensions o f a packaged microwave am plifier.
The package is modeled as a PEC box w ith two holes at the in p u t/o u tp u t ports
for feeding power.
54
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5
0
-5
-10
-15
4
8
12
16
20
Frequency (GHz)
F ig u re 5.2: The F D T D sim u la tio n o f a packaged u n ifo rm m ic ro s trip lin
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
7.5
\
3
5.5
O
- - w/o package
w/ package
4.5
6000
6500
7000
7500
8000
Time Steps
Figure 5.3: A n exam ple showing the effect o f the package on the c irc u it response.
the resonance frequency come from the num erical error clue to ta k in g F F T on
the late-tim e response w ith out sufficient tim e sequences.
F D T D s im u la tio n indicates th a t the packaged am p lifie r is stable.
Fig. 5.3
shows the c irc u it response at steady state. The package affects the c irc u it so that
the m agnitude o f the observed voltage at the o u tp u t p o rt becomes larger when the
circu it is placed in the package. Fig. 5.4 shows the effect o f the packaging structure
on S-parameters. In this case, the frequency o f the m atching d ip rem ains the same
at 5.58 GHz b u t the m agnitude changes from -8.98 dB to -10.07 d B . while the
gain at 6 GHz varies from 10.95 dB to 11.76 dB. The packaging effect on the
o u tp u t power o f a single-tone excitatio n at 6 GHz is shown in Fig. 5.5.
The
resonant frequency is close to th a t of the second harm onic, so energy is coupled
to the second harm onic strongly and its power curve varies m ore significantly
than others.
56
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
20.0
— w/packaging
- w/o packaging
10.0
0.0
-
-
10.0
20.0
2.0
4.0
8.0
6.0
10.0
Freq (GHz)
Figure 5.4: Investigatio n o f the packaging effect on the sm all-signal analysis by
e xcita tio n o f a m odulated Gaussian pulse.
50.0
-x w/ packaging
w/o packaging
fundamental
0.0
3rd
2nd
-50.0
-
100.0
-10
0
10
20
P„ (dBm)
Figure 5.5: The packaging effect on the large-signal analysis by a single-tone
excitatio n at 6 GHz.
57
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5.3
O scillation
T h is section discuss the o scilla tio n phenomena. From the v ie w p o in t o f c irc u it
theory, the interaction between the package and the c irc u it means th a t there is
a sig nal p a th from the o u tp u t p o rt to the input p o rt th ro u g h the interactio n.
E sse n tia lly i t is a signal feedback affe cting the c irc u it perform ance.
Once the
feedback is very large, it m ight force th e c irc u it to be unstable. P h ysica lly speak­
ing, th e o u tp u t power is coupled to the resonance mode o f th e package. Under
such a s itu a tio n , the active device generates added power to th is feedback power
and re ta in s an oscillation, alth o u g h th e package is semi-open. T h is phenomena
can o n ly be observed in nonlinear analysis o f the entire system.
T he dim ensions of a larger packaging box are chosen to be 1560 m ils x 186 m ils
x 1250 m ils. B y F D T D pre -sim u la tio n o f a packaged u n ifo rm m ic ro s trip line, the
firs t resonant frequency o f this packaging structure is found at 5.72 G H z. Using a
sm a ll-sig n a l Gaussian pulse m odulated a t 6 GHz, close to the resonant frequency,
the o u tp u t voltage is as shown in Fig. 5.6. The o u tp u t voltage o f the packaged
c irc u it in Section 5.2 is also p lo tte d in the same figure. As a stable c irc u it, only
the m o d u la te d Gaussian pulse appears at the output p o rt. b u t. as can be seen,
there is an oscillation after signal e x c ita tio n . Fig. 5.7 shows the spectrum o f the
o s c illa tio n . The resonance is at the same frequency o f the firs t resonance mode,
w hich means th a t the packaging stru c tu re interacts w ith the c irc u it heavily and
the c irc u it becomes oscillating.
58
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10
Small package
Large package
8
~o
>
a)
cn
aj
o
>
-♦—*
13
6
Signal
4
Q.
-4—*
o
2
0
2000
4000
6000
8000
Time steps
Figure 5.6:
T he o u tp u t voltage o f the a m p lifie r in the cases o f two d iffe re n t
package dimensions.
59
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3
6
9
12
15
18
Freq (GHz)
Figure 5.7: The spectrum o f the o u tp u t power in the case th a t a sm all-signal
Gaussian pulse m odulated at 6 GHz is imposed upon a packaged am plifier; this
example dem onstrates c irc u it oscillations caused by the packaging structure.
60
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CHAPTER 6
CROSSTALK IN MULTICHIP MODULES
6.1
Introduction
T h is chapter discusses issues o f electrom agnetic interferences in a m u ltich ip m od­
ule.
In M M IC -based systems, such as microwave transceiver modules, several
active circu its are integrated in a module and enclosed in a m etal package to
protect the system fro m external electrom agnetic interference. As the operating
frequency scales up to the m illim eter-w ave spectrum, these c irc u its unavoidably
encounter in terna l electrom agnetic interference, in c lu d in g the effects of the mu­
tu a l coupling between closely spaced circuits and the in te ra ctio n between the
c ircu its and the m etal package. The packaging effect is discussed in C hapter 5.
T h is chapter focuses on the m u tu a l coupling effect.
In a m icrostrip c irc u it, discontinuous structures such as m atching networks
cause ra d ia tio n o f power, p a rtic u la rly in high frequency o f operation. Enclosed
in the package, this power is coupled to other circuits. Also, power may leak to
c ircu its in close p ro x im ity thro ugh substrate coupling. T he coupled power essen­
tia lly becomes crosstalk noise to neighboring circuits. A n engineer is concerned
w ith the electrical iso latio n between circuits in system design.
C rosstalk in microwave passive circuits has been extensively studied in l i t ­
erature. Much a tte n tio n has been focused on characterization o f crosstalk phe­
nomenon between three-dim ensional passive structures, such as bonding wires
61
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Three-dimensional
passive structures
Nonlinear
circuit
Electromagnetic
sim ulator
M ulti-port
network
Equivalent
sources
Network
simulator
Crosstalk
analysis
Figure 6.1: F lo w chart of conventional procedures for crosstalk analysis.
[31] and m ic ro s trip lines [32, 33], or spurious coupling between d iffe re n t ports
o f a package [34, 35]. Furthermore, different approaches have been proposed to
reduce crosstalk. R e duction of the coupling can be achieved by b u rie d structures
[36], by separators [37], or by inhomogeneous substrate [38]. To date, crosstalk
analysis in active c irc u its is constrained by sim u la tio n tools. W h ile a va rie ty o f
full-wave techniques have been developed, m ost o f them s till follow the conven­
tional procedure o f c irc u it sim ulation. The conventional procedure as discussed
in [39] is illu s tra te d in Fig. 6.1. The passive stru ctu re is analyzed by electrom ag­
netic sim ulators in order to perform param eter extraction o f its n e tw o rk model,
e.g. reference [40]. N onlinear active circu its are modeled as equivalent sources.
62
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Three-dimensional
passive structures
Nonlinear
devices
Extended
electromagnetic
simulator
Comprehensive
simulation
Figure 6.2: F lo w chart o f comprehensive s im u la tio n by an extended electrom ag­
netic sim u la to r.
Crosstalk analysis is then performed by c irc u it sim ulato rs using the sources as
power generators in the m u lti-p o rt network. W it h an extended electrom agnetic
sim ulator, the procedure can be realized in a com prehensive sim ulation, as shown
in Fig. 6.2. T h e configuration o f a m u lti-c irc u it m odule is described in Section 6.2.
In Section 6.3, the extended F D T D m ethod is a p p lie d to perform crosstalk anal­
ysis. R e duction o f crosstalk by metal separators is discussed in Section 6.4.
6.2
M od elin g o f the M odule
S im ila r to the co n fig u ra tio n for analysis o f packaging effect in Fig. 5.1, the config­
u ration o f a m u lti-c irc u it module is illu stra te d in Fig. 6.3. where two microwave
am plifiers are enclosed in a packaged module. B o th am plifiers are identical to the
one described in C h apter 4. The center-to-center distance between the am plifiers
63
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Packging Box
irt 2
Vdd
Circuit I
Vdd
port 1
port 4
port 3
Circuit 2
Figure 6.3: The configuration of a tw o -a m p lifie r module.
is m arked as D . The package is sim ply modeled as a perfect electric conduc­
to r ( PE C ) box o f zero thickness w ith holes, each in the size o f 19 x 10 cells
in the direction o f w id th and height, respectively. In the sim ulation, there are
4 F D T D cells, or 40 m ils, between the package side w a ll and the AB C plane.
Power feeding is modeled in a sim ple manner. There are four ports.
A t each
p o rt, a m icrostrip line passes thro ugh the hole on the package and connects to
a resistive power source. T h e substrate extends outside the package to support
the m icro strip line. The power sources, each w ith a source impedance o f 50 fl.
provide DC biasing conditions by setting Vgg and Vdd to the necessary values and
A C power by setting Pin. As the distance D changes for diffe re t analysis, the
po sitio n o f holes also varies so th a t the m icrostrip line is located a t the center o f
holes.
64
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Figure 6.4: A snapshot o f field d is trib u tio n at the tim e step 6000.
6.3
Crosstalk A nalysis
T h is section discusses crosstalk phenomena in the nonlinear system.
T h e ex­
tended F D T D m ethod is applied to perform large-signal analysis o f the entire
system. As such, the packaging effect is also included in the sim u la tio n .
The
dim ensions of the package is chosen as 1280 m ils x 186 m ils x 690 m ils.
By
F D T D pre-sim ulation, the firs t resonance frequency is found at 9.1 G Hz. B o th
am plifiers are biased at the same o p e ra ting p o in t, Vgg = -0.81 V and I
AC power o f a sinusoidal signal a t
6
G H z is pumped into p o rt
inspects the coupled noise power in c irc u it
2
1.
m
= 6.4 V'.
T h is analysis
.
Fig. 6.4 shows a snapshot o f steady-state field d istrib u tio n d u rin g s im u la tio n
a t the 6000th tim e step. The field d is trib u tio n at circu it 2 appears fla t because
the c irc u it is only DC biased. Ideally, there is no AC power at the in p u t/o u tp u t
p o rts o f c irc u it 2 i f electrical iso la tio n between the two circuits is good. However,
the field d is trib u tio n between tw o circu its near the closely spaced m a tch in g stubs
65
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is n o t fla t. T h is indicates the presence o f crosstalk. Fig. 6.5 shows the near-end
and far-end coupled noise in tim e domain. D iffe re n t curves show the responses
by va ryin g the pum ped power level. As the in p u t power level increases, ha rm o n ic
power also increases. Thus the voltage curve does not appear sinusoidal. T h e
spectra o f these curves are obtained by ta king the Fourier transform o f the tim e
responses. Results o f the far-end noise are shown in Fig.
6 .6
. I t is evident th a t
harm onics generate a t 12 and 18 GHz. The coupled power o f harmonics comes
from two d ifferent sources. One is d ire ctly coupled noise a t the o u tp u t p o rt: the
other is a m p lifie d power o f the coupled noise a t the in p u t p o rt. The peak around
9 GHz is essentially the resonance frequency the package.
A ccording to the crosstalk analysis, Fig. 6.7 shows the curves o f the coupled
noise power to the in p u t power at
6
GHz.
Design curves can be obtained by
varying different param eters. For example, one param eter is the center-to-center
distance between circu its. Three different cases have been tested. Results are
shown in the same figure.
For the case th a t D equals to 470 mils, i t is o n ly
20 m ils between the two m atching stubs at the in p u t p o rt. As the in p u t power
increases, the coupled noise also increases. The o u tp u t coupled noise appears to
be a linea r fu n c tio n o f the in p u t power, in dica ting th a t c irc u it
operating region. Besides, the output power o f c irc u it
varies. Fig.
6 .8
1
2
is s till in the linear
rem ains the same as D
shows the curves of the coupled noise power to the distance D .
Results o f choosing d iffe re n t in p u t power levels are p lo tte d . T he coupled power
decreases as the distance increases, sim ila r to 1 /r-ru le .
The calculations show
th a t the m u tu a l co u p lin g can be very high up to the order o f -10 dB. Depending
on the range o f o p e ra tin g power, the separation between c irc u its should be chosen
carefully.
Broadband analysis o f crosstalk is realized by sm all-signal analysis. Fig. 6.9
66
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-0.6
-
-0.7
Pin = 5.95 dBm
Pin = 9.93 dBm
Pin = 13.91 dBm
Pin = 15.96 dBm
CO
75
>
-
0.8
-0.9
-
1.0
10000
10500
11000
Time Steps
11500
12000
(a)
6.4
Pin = 5.95 dBm
Pin = 9.93 dBm
Pin = 13.91 dBm
Pin = 15.96 dBm
6.3
6.2
CO
o 6.1
>
6.0
5.9
5.8
10000
10500
11000
Time Steps
11500
12000
(b)
F ig u re 6.5: The coupled noise voltage at (a) the near end and (b) the far end by
v a ry in g the input power level a t the input p o rt o f the c irc u it
1.
67
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3rd
2nd
-20
-40
7' » nv
i i
E
-60
CD
"O
-80
Pin = 5.95 dBm
Pin = 9.93 dBm
Pin = 13.91 dBm
Pin = 15.96 dBm
-100
-120
20
Frequency (GHz)
Figure
6 .6
: The spectrum o f the far-end noise power in Fig. 6.5(b).
circuit 1 \
E
CD
73
3
o
circuit 2
CL
-10
D = 470 mil
D = 490 mil
D = 530 mil
-20 f
-30
-5
0
5
10
Pin (dBm)
15
Figure 6.7: The coupled noise at the o u tp u t port o f c irc u it
20
2
by va ryin g the in p u t
power level.
68
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10
j - ---------— —,--------------------------------------------- Ii
— Pin = 9.9 dBm
- Pin = 13.9 dBm
E
m
■o
o
<D
■C
w
-10
3
O
0.
-20
460
Figure
6 .8
480
500
520
540
560
D (center-to-center) (mil)
580
: T h e coupled noise at the o u tp u t p o rt of c irc u it
600
2
by varying the
center-to-center distance D.
and 6.10 illu s tra te S-parameters o f th is fo u r-p o rt system.
The S-parameters
of a single a m p lifie r w ith the package is also p lo tte d in Fig. 6.9 fo r com parison.
The discrepancy comes from the packaging effect. Case studies by varying the
distance D are tested. Tw o different values o f the distance D have been tested.
The S-parameters o f c irc u it
circu it
at
6
2
decrease by
8
1
do not change much, while the S-param eters o f
dB as the distance D changes from 470 m ils to 530 m ils
GHz. C rosstalk between these two c irc u its is very sensitive to the distance
D.
6.4
R ed u ction o f Crosstalk B y Separators
An im p o rta n t issue in electromagnetic interference is reducing crosstalk.
The
coupling th ro u g h radiative interaction can be reduced using resistive film on the
69
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2
20
— D = 470 mil
- - D = 530 mil
Single amplifier
0
•2
4
■6
■8
-10
-1 2
-1 0
Frequency (GHz)
Figure 6.9: The S-param eters o f c irc u it
1.
-10
-20
CO
-o
-30
-40
D = 470 mil
D = 530 mil
-50
4
6
5
7
8
and c irc u it
2
Frequency (GHz)
F igure
6 .1 0 :
The coupling between c irc u it
1
.
70
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-10
-20
-40
-50
— w/o separators
— Case a: metal pins
■- Case b: metal plate
4
6
5
7
8
Frequency (GHz)
Figure
6 .1 1 :
T h e S-parameters o f c irc u it 2 show ing the reduction o f crosstalk.
package w all to absorb ra d ia tio n energy. The analysis was realized by the F D T D
method in reference [27], T h is section discusses the substrate coupling.
The stru c tu re o f m ic ro s trip circuits seldom generates surface waves in th e
substrate.
B u t A C power can be coupled to neighboring elements i f they are
placed too closely. As such, separators can be placed between circu its in ord e r
to block the co up ling. N ote th a t the separator should o n ly reduce the crosstalk
w ith o u t affecting the performance o f in d iv id u a l circuits. In reference [41], lossy
vias are placed in a ceram ic package to suppress the package resonance. In th is
analysis, m etal pins are used. Fig.
placing separators.
6 .1 1
show th e S-parameters o f c irc u it
There are 13 pins placed in the substrate.
2
a fte r
The distance
between two neig h b o rin g ones is 50 mils. Results o f the extreme case, placin g
a m etal plate in the substrate, are also shown.
decreases by 3 d B at
6
As can be seen, the coup ling
G Hz in the case of using m etal pins and 7 dB in the
71
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20
m
73
CO
w/o separators
Case a: metal pins
Case b: metal plate
-10
-10
-12
Frequency (GHz)
Figure
6 .1 2 :
The S-param eters o f circu it
1
a fte r p lacin g separators.
extrem e case. These separators have no significant effect on in d iv id u a l circuit
responses, as one can see in F ig. 6.12.
72
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CHAPTER 7
ABC REALIZATION B Y DIGITAL FILTERS
7.1
In trod u ction
In applying th e F D T D m ethod to the analysis o f waveguiding structures, the
com putation do m a in is truncated because o f lim ite d com puter m em ory. Thus,
the dom ain should be as small as possible to decrease com putation tim e . Conse­
quently, a special a lg o rith m , known as the absorbing boundary c o n d itio n ( A B C ),
is required on the trun cate d boundary in order to reduce spurious reflection from
the boundary. A good A B C performs well wave absorption over a w ide frequency
range w ith o u t m uch com puting cost.
M any m ethods o f A B C realization have been developed.
One p o p u la r ap­
proach, based on M u r’s method [42], applies cascaded one-wave equations by
recognizing th a t the fields near the boundary are outgoing in the w aveguiding
direction [43. 44]. To each one-wave equation, there is a constant chosen corre­
sponding to th e wave phase velocity at a certain frequency such th a t the A B C
can absorb waves at this frequency. Since wave velocities can o n ly be m atched
at some discrete frequencies, wideband absorption cannot be easily obtained in
highly dispersive structures. A different approach places layers o f non-physical
absorbers, called the perfectly matched layers ( P M L ). su rro unding the com ­
putation dom ain [45, 46]. This A B C realization can absorb p ro pa gating waves
in a wide frequency range in dispersive structures.
Furtherm ore, an extension
73
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has been derived to adequately absorb evanescent waves [47]. The c o m p u tin g
effo rt, however, increases d ra m a tic a lly as the num ber o f PM L cells increases such
th a t the num ber is suggested to be fro m 4 to
8
[48]. Another approach u tiliz e s
the d iakoptic technique, which applies a tim e -d o m a in Green’s fu n ctio n o b ta in e d
by pre-sim ulation [49], or by th e o re tica l fo rm u la tio n [50], Usually th is approach
needs tim e-consum ing calculation o f tim e convolution, unless a fast recursive al­
g o rith m is employed, e.g. the Laguerre discrete-tim e network used in [51]. A n
a lte rn a tive approach realizes the A B C based on the technique o f d ig ita l filte r
design [52], in which an allpass filte r has been designed to absorb p ro p a g a tin g
waves in an a ir-fille d waveguide.
D ig ita l-filte r realization provides w ideband absorption and takes lit t le com ­
p u tin g effo rt as w ell as m em ory burden.
In th is Chapter, the d ig ita l-filte r ap­
proach is generalized to absorb p ro p a g a tin g and evanescent waves by using a bank
o f in fin ite im pulse response (IIR ) filte rs . T h e o re tica l form ulation is discussed in
Section 7.2.
The absorption o f m u lti-m o d e s can also be achieved by cascad­
in g several filters w ith out cumbersome mode e xtra ctio n . Different w aveguiding
structures in clu d in g rectangular waveguide and m icro strip line are tested in Sec­
tio n 7.3. In Section 7.4, an iris s tru c tu re is used to demonstate the perform ance
o f the approach.
7.2
Theoretical Form ulation
Consider an A B C boundary where waves are outgoing in the + z d ire ctio n . F ie ld
u p d a tin g on the boundary does not fo llo w the norm al F D TD equations in Sec­
tio n 2.1 because exterior fields are in v a lid . Since field components near the b o u n d -
74
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ary satisfy the one-wave equation.
dE
dE
3F + F 9 i ~ 0'
( , 1)
the is-field can be expressed as
E {z ) = E 0ej ^ l - l( “ )z\
where
7
(7.2)
, equal to 0 — j a . denotes the com plex propagation constant a n d is
frequency dependent.
Therefore, the value o f the tangential field E m on the
boundary can be updated from the value o f the in te rio r field E m -
i
according to
(7.3)
E m- l
where the exponential facto r indicates the p ro p e rty o f wave propagation, in c lu d ­
ing propagating and evanescent waves.
The basic idea o f th is fileter-like approach is to tran form Eq. (7.3) in to a
finite difference form , or equivalently, a discrete-tim e system, in which the system
function is defined in the frequency band o f interest as
G (( l) = —y °
h p~ina
--
■ = e- i r <n>a=
( 7..n
where Q is the norm alized frequency. A c tu a lly , this discrete-tim e system is an
iVth-order H R filte r. The in p u t and o u tp u t signal are E m - \ and E m - respectively.
The concept is illu s tra te d in Fig. 7.1. The same H R filte r is applicable to u p d a te
fields over the entire A B C boundary, since the phase o f the guided wave follow s
Eq. (7.2) not on ly in a certain grid b u t also over the entire boundary.
In ideal cases T(f2) should equal to
7
(0 /), b u t, otherwise, a reflected wave is
generated from the a rtific ia l boundary. The reflection coefficient can be derived
from the continuity- co n d itio n on the A B C boundary.
The to ta l fields on the
75
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ABC plane
Az
M -2
M-l
V
A
G(cd)
v
A
Leapfrog scheme ■
M
A
Digital Filter
F igure 7.1: The fields on the boundary are determ ined by the value of in te rio r
fields, w hich is represented by a d ig ita l filter.
boundary' ( z = 0 ) and in the in te rio r point ( c = —A ;: ) are
E x t-i
=
E 0elAz + /2£'0 e_7A*
at c = —A c
E \t
=
(1 + R )E 0
at c = 0
(7.5)
where R is the reflection coefficient. A fter m a n ip ula tion. R is derived as
D_
_ - j( r - 7 )Ac _ 1
__________ *
(7.6)
1 _ e-j(r+7)As •
T h is equation is used as the goal function to design the filte r. The filte r coeffi­
cients an and bn are o p tim ize d to m inim ize the discrepencv between Eqs. (7.3)
and (7.4) over a wide frequency range,
m in ^ 2 w k
k
V
N b p~]nu,k
2 - n = 0 un L
Y.n= 0 O n e - i ™
(7.7)
*
where w k is a weighting fu n ctio n . The procedure to solve fo r an and bn is based
on the least-square-error a lg o rith m in [53]. Note th a t a ll o f the calculations are
perform ed in the frequency dom ain. To ensure the scheme is stable, the poles
o f the filte r must be w ith in the u n it circle in the c-dom ain.
The structure of
the discrete-tim e svtem is realized by cascade-form or p a ra lle l-fo rm structures so
th a t the effect o f q u a n tiza tio n errors is reduced [54].
To absorb m ulti-m o de waves, each mode requires a filte r since the propagation
constant o f each mode varies. In doing so, the m agnitude o f each mode needs
76
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G1
G2
Em
G2
G1
F igure 7.2: The A B C config uration o f the two-mode case by a p p ly in g a bank o f
filters.
to be extracted from the to ta l field a t the position o f M — 1 . T yp ica lly, mode
e x tra c tio n is quite cumbersome and requires prior knowledge o f m ode patterns.
Nevertheless, mode e xtra ctio n can be avoided by in volving m ore in te rio r fields in
the calculation. For exam ple, in the case o f two-mode abso rp tion, the field Em
on the boundary is evaluated from in te rio r fields E m - l at c = —A c and E m - 2
at c = —2Ac. The to ta l fields at different positions are decomposed in to the two
modes as
E m —2 =
<
E ‘2 i + E 2 2
-Em
where G \ and
at c = —2 A c
E \\ + E \ 2
= E 0 1 G 1 + EoiG'i
at z -- —A c
— Eqi + Eq2
= E-2 iG f + E-riGk
at c =
(< -8 )
0
denote the filters fo r the two modes. The field E_\[, therefore,
can be derivded and expressed in term s o f £ a / - i and E m - 2 as
E m = E \ [ - i ( G i + Go) — E m - zG i G o.
(7.9)
T he configuration of the filte r for the two-mode case is depicted in Fig. 7.2. Hence,
d ire ct calculation o f E m is achieved. The form ulation can be generalized to the
case o f m ulti-m odes and a bank o f H R filters are cascaded fo r im plem entation.
(t
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
7.3
A b sorp tion of Single M od es
The approach is firs t used to sim ulate waveguiding structures to exam ine its
performance. The firs t case is the a b so rp tio n o f the dom inant m ode in an em p ty
rectangular waveguide, which is a h ig h ly dispersive structure. T h e cross-section
dimensions o f the waveguide are 3 cm x 1.5 cm, divided in to F D T D cells o f 30
x 5. The space step A z chosen is 0.5 m m .
A second-order IIR filte r is chosen fo r im plem entation.
To o p tim ize the
coefficients o f the filte r, a very long waveguide is pre-sim ulated to o b ta in the
complex pro p a g a tio n constant o f T E \ q mode, which comes fro m the ra tio o f
recorded E -fie ld s at two neighboring p o ints. The phase constant is norm alized
to th a t o f waves in free space and p lo tte d in Fig. 7.3.
Note th a t the curve
shows the frequency dispersive characteristic. The frequency range is above the
cutoff frequency a t 5 GHz. Subsequently, the coefficients are determ ined as
= -1.4074665, a2 = 0.40844664, b0 = -0.35247524. bx = 1.2957203. and b2 = 0.94269684. T h e optim ized curves are also p lo tte d in the same figure. The poles
of the filte r are at zt = 0.99834113 and z2 = 0.40912533. B oth are inside the
u n it circle. T he reflection coefficient calcula ted by F D T D s im u la tio n is p lo tte d
in Fig. 7.4, illu s tra tin g that this A B C re a liz a tio n absorbs better th a n 45 dB in a
wide frequency range. I f a third-ord er filte r is used, the absorption increases up to
60 dB. Results o f F D T D calculation are in good agreement w ith calcula ted data
according to Eq. (7.6), indicating th a t the performance of an A B C re a liza tio n is
available in advance provided th a t the com plex propagating constant is known.
The second case is the absorption o f evanescent waves. Using th e same wave­
guide, p re -sim u la tio n is performed by e xc ita tio n of a single T E 20 mode in an
empty waveguide. The frequency range is below the cu to ff frequency. 10 GHz.
The norm alized attenuation constant is shown in Fig. 7.5. Since an evanescent
78
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- - Pre-simulation
— MR filter (N=2)
£ 0.6
0.4
Frequency (GHz)
Figure 7.3: T h e norm alized phase constant o f a recta ngular wageguide.
The
system fu n ctio n o f a second-order H R filte r is designed to m atch the pre-sim ulated
data.
0
FDTD
Formula calculation
-20
-40
N=2
-60
N=3
-80
-100
5
20
10
15
Frequency (GHz)
Figure 7.4: C a lcu la te d reflection coefficients o f F D T D sim u la tio n and num erical
theory based on Eq. (7.6) using an IIR filte r o f order
2
and 3.
79
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_ 4
c
CO
GO
c
Pre-simulation
MR filter (N=2)
° 3
Q
o
c
o
cc
3
c
<u 2
1
0
2
4
6
8
Frequency (GHz)
10
F ig u re 7.5: A second-order H R filte r is optim ized to m atch the norm alized a t­
te n u a tio n constant o f a recta ngular wageguide.
wave decays as it travels, the reference plane must be very close to the A B C
plane to be able to observe the evanescent wave. In this sim u la tio n , the distance
between the reference and A B C plane is 5 F D T D cells. Fig. 7.6 shows the cal­
cu la te d reflection coefficient. Results o f placing a PEC on the boundary is also
p lo tte d in the same figure as a reference showing the order o f m agnitude by to ta l
reflection.
T he approach is also applicable to open structures, such as a m icro strip line.
T h e thickness of the substrate, w ith a dielectric constant o f 9.9. is 10 m ils. The
w id th o f the m icrostrip line is chosen as 9.6 m ils.
The com plex propagation
constant is obtained by a p re -sim u la tio n of dom inant-m ode e xcitatio n.
coefficients o f an optim ized second-order HR filte r are
0.98423582,
6o
= -0.95873958.
6
The
= -1.9805548, a2 =
t = -1.9185786, and b2 = 0.96352007. The po-
80
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-20
— FDTD
- Formula calculation
PEC plane
CQ
■a
cc
-100
3
6
Frequency (GHz)
9
Figure 7.6: Results o f evanescent-wave absorption. The results o f p la cin g a PEC
on the bo u n d a ry shows the effect o f to ta l reflection for comparisons.
sitions o f poles are at zt = 0.99027738 + jO.059887608 and z2 = 0.99027738jO.059887608. Since the m icro strip lin e is not a highly dispersive s tru c tu re , the
reflection coefficient can be reduced to less than -60 dB over a w ide frequency
range fro m 0 to 20 GHz as shown in Fig. 7.7.
7.4
A bsorption of M u lti-m od es
A b so rp tio n o f m u ltip le modes is studied using the structure o f an in d u c tiv e iris,
w ith gap w id th equal to 0.6 inch, inside a W R -90 waveguide as shown in Fig. 7.8.
The space steps are A x = 0.762m m , A y = 2.54 mm and A x = 0.5 m m .
A
m odulated Gaussian pulse centered a t 10 G H z is used as the source. The iris
generates high-order T E n0 modes. Unless the distance between the d is c o n tin u ity
and the A B C plane is far enough such th a t evanescent waves decay to be ne-
81
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-20
FDTD
Formula calculation
-40
CQ
-o
CE
-60
-80
-100
10
15
5
Frequency (GHz)
0
20
Figure 7.7: T h e ca lcu la tio n of reflection coefficient fo r a m icro strip lin e . A sec­
ond-order H R is used in the sim ulation.
ABC plane
2.5 mm
0.4 in
0.6 in
h
0.9 in
Figure 7.8: The stru ctu re o f the inductive iris.
82
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* Long W aveguide
One Mode ( 0.1 X)
- Two Modes ( 0.1 X )
- Three Modes (0 .1 ? .)
One Mode ( 0.2X)
-
co
-13
-15
8
9
10
11
12
Freq (G Hz)
F igure 7.9: The calculated reflection coefficient o f an in d u c tiv e iris in a W R-90
waveguide. The A B C plane is placed at the distance o f 0.2 A and 0.1 A from the
iris. In the la tte r case, variours m ulti-m ode absorption for T E i 0, T £Ao- and T
0
modes are tested. T h e results are compared w ith those o f a long waveguide.
glectable, these high-order modes affect the S-param eter calculation. Results o f
F D T D sim ulations are shown in Fig. 7.9. Results o f a very long waveguide is
used fo r comparisons. Several cases has been tested. I t is shown th a t the A B C
plane should be placed at 0.2 Acarrier, or
12
F D T D cells, fro m the iris in the
case th a t single-mode abso rp tion is applied. I f the A B C plane is placed closer to
the iris at the distance o f 0.1 Acarner, absorption o f high-order modes should be
considered. In this s itu a tio n , three-mode absorption, in c lu d in g T E w ,T E -2q and
T E m modes, provides the best absorption. In a d d itio n , those curves o f different
cases also provide in fo rm a tio n about how each mode affects on the S-parameters.
83
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CHAPTER 8
CONCLUSIONS
Extended electrom agnetic sim ulators are the solution fo r system design o f n on lin­
ear m icrowave integrated circu its in a com plicated enviro nm ent. As the system
is operated at a high frequency, electromagnetic effects are no longer neglectable.
Reliable analysis m ust include these effects to achieve comprehensive sim ulation
o f the entire system.
In th is dissertation, electrom agnetic sim ulation o f packaged microwave active
circu its is realized by extending the F D T D m ethod to go beyond the ca p a b ility
o f c irc u it sim ulators. The circu its include three-dim ensional discontinuous struc­
tures and nonlinear active devices. The conventional F D T D a lg o rith m is extended
to incorporate active devices by using equivalent sources. These sources charac­
terize the voltage-current relationship as well as the sca tte rin g properties o f the
active device. Being su b stitu te d by equivalent sources in the device region, the
active device is represented by its lumped c irc u it m odel w ith the spatial effect
accounted for. A n equivalent c irc u it is accordingly derived to characterize device­
wave interaction. A n ite ra tive scheme is applied to evaluate the device voltage
and update the electrom agnetic fields in the device region. T he m odeling error
o f th is approach is n u m erica lly estimated by testing a sim ple configuration o f a
m icro strip line term in ate d into a matched resistive load.
T h is approach is general and applicable for m u lti-p o rt devices. It is applied to
sm all-signal analysis o f a M E S F E T microwave a m p lifie r. C alculated results are
84
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valida ted by comparisons w ith measured data in good agreement. Large-signal
analysis fo r characterization o f non linear properties demonstrates the a p p lic a tio n
o f the approach fo r practical analysis and design. The approach is then employed
to investigate electrom agnetic interferences o f packaging effect and crosstalk. In i­
t ia lly adopted to protect the system from external interferences, the packaging
stru ctu re m ay affect circu it perform ance due to radiative in te ra ctio n and may
cause o scillations in certain cases. C rosstalk due to near-field coupling is studied
using a m u lti-c irc u it module. I t is shown th a t the far-end coupling can reach for
-10 dB . Design curves are obtained th ro u g h comprehensive sim ulations. Cases
are discussed to reduce substrate co u p lin g by placing m etal separators between
circuits.
A n absorbing boundary co n d itio n fo r the F D T D m ethod is realized using a
discrete-tim e system by the technique o f d ig ita l filte r design. T his approach is
generalized to be able to absorb p ro pa gating and evanescent waves over a wide
frequency range. A bsorption o f m u ltip le modes is achieved by using a bank of
d ig ita l filte rs to avoid cumbersome mode extra ction. The approach is firs t applied
to several tests o f a rectangular waveguide and a m icrostrip line, and results show
very good absorption. A pplied to an iris problem , the realization perform s so well
th a t the A B C plane can be placed very close to the discontinuity, o n ly 0.1 A or
6
F D T D cells, w ith o u t significantly influencing the calculation. N u m e rica l cost,
therefore, can be greatly reduced.
In general, full-w ave analysis is applicable to packaged nonlinear active in­
tegrated circ u its .
A lth ough full-wave sim ulators are s till more tim e-consum ing
as com pared to c irc u it sim ulators, th is analysis becomes necessary and provides
useful in fo rm a tio n for circuit design in the environm ent where electrom agnetic
effects o f ra d ia tio n and coupling m ust be considered. Moreover, electrom agnetic
85
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sim ulation can provide visu alization ca p a b ility o f field propagation.
The applications discussed in this study is emphasized m ainly on in te rn a l elec­
trom agnetic interferences. There are also ap p lica tio n s for exterior interferences,
for instance, interference im m u n ity of electronic equipm ent, which is rig o ro u sly
regulated by FCC. In the area o f wireless com m u nica tion, it is a c ritic a l problem
to design a handset antenna w ith diversity and low absorption in the hum an head
[55, 56]. The study o f these issues by the F D T D m ethod is receiving more and
more interests. In the future, comprehensive s im u la tio n o f the entire system by
extended electrom agetic sim ulators may become feasible and im portant.
86
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R
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