SPRIM: Structure-Preserving Reduced-Order ... - Semantic Scholar

SPRIM: Structure-Preserving Reduced-Order Interconnect Macromodeling Roland W. Freund

Department of Mathematics University of California, Davis One Shields Avenue Davis, CA 95616, U.S.A.

[email protected] ABSTRACT 

   !
 "
$#&%'()(
)*+, .-

Y
_
) .^@(\1, 
9[ M"
$# %(

%/(&"'-  %0!12 "31 "4(56"($758(:9!
 
9

!
mP4!:;  //u 1)  !
Q"'- f%0
c(9!
$dQ \%0B/0J  5{,i  (,$^5{+-( !S?? ,$  !
N bG }  " e€U ^"

1)!1, ;(5,? ?@/0( BA/(/0  :50 "3$9 
',i?? G:
O„?†.Œi†Ž!‡? ^ ^"J7
= "'. SkTNP&l58(V9(
,RA /  `(B "\%,  b U M >5 ">? V1).  ]
"M
+ aASChD– ? G } " e"(e1(  -( X ">-!, !/1
 U†.ƒ7J7? "I@…!ƒM]
'

sS1 ,',? 3.S/0  "'  ,$, U  < "M $(
i&J2a% ›O-$MJ ")5{1œ5, 
,?y/0
'
B(
`,  



È 3¾(ÄÈ =%,  ba*+, .-\#
 ï  ÓiÔ ÿ  « ÿ ¨«««¨þ ( ¨þ

!, 1
J e!% 
= " ÿ Ô

¨þ

ÿ

ï

søEV ª6¨aÙRV ª)¦_ø V

Ö

øiª]¨

ª£

Ù^«

€,  9!" ,?O1>9!
,:? '.  !
[5 }  " (1™… G

è

ƒMjG !G 

Ci

1b2c4:578469œ… G G Ä8ÁÈ
&I ½ÃÄ#k=;Âf=)ÈM¿ÄjÃ=Å{¿xÄ8ÁÈmÈ@Mʾ(¿ÃÅÂ(¿Ã ÂO B ãg¾(¿d;

Ä8ÁÈ3ʽÂ.l ÈÀÄÈ ; = Â(

ÈÉ ã:V ªmh

È &I ¿0Èh\

¼  U†‰ ƒe¾(¿8

@…Š ƒƒ ¾!¼ÂÆÄ G Ä8ÁÈ
&I ½ÃÄa….#R; = Âf) = ÈM¿ÄjÃ=Å{¿xÄ8ÁÈmÈM@ ʾ(¿ÃÅÂ(¿à ÂO B ãg¾(¿d;

Ä8ÁÈ3ʽÂ.l ÈÀÄÈ ; = Â(

ÈÉ ã:V ªmh

È &I ¿0Èh\

¼  U†‰ ƒe¾(¿8

@…Š ƒƒ ¾!¼ÂÆÄ
x%0=J3?  
_$
q(

−2

10

(, , N(
'@75{1NG −3

ž_ä]Øiƒ ÓEÔ

"e1) ?uc Ö


[@‹!ƒ2 7(5h "+5{1

žIä]Øiƒ ÓiÔ ¦

 Ö

ª

¬«

 =@ä.ƒƒ ÓEÔ Ú

«

«

¨ Ü

!Œ!ƒ

abs(Z(2,1))

10


'](
9S " %,  MbVU  Sˆ ƒM ? R R(UV S- ?58V "'.

−4

10

−5

10

−6

10

J7"

ªN@ä.ƒR¦zä4Ï Ô

ž_Î

† ž

Ô

Î

ä

£ï

ï «

ï ÓEÔ Î Ï

( ƒ


 
9Q!Œ!ƒ7$
c@‹!ƒM6? S5{!, , J7+ "'. < "e 
U5{M+5{
  !
ã_(
X,$%0SJ3?  
X(75{(, ,$J7¢ ã^@ä.ƒR¦

£ ð

„ Ô

ähÏ

žqÎ Ô

Ô

† ž

£ïfÏ Î

ä

… ï

ï ÓiÔ Î

ÓiÔ ð

­

« Ô

−8

10

0

cª

0.5

1

1.5

2

2.5 3 Frequency (Hz)

±7¤Cw8²³y´¶µd}¢ · sï ¸ Ô ¢8¹xdy p

!‹!ƒ

T=.- !$
f- J`(5E‰ ƒM ":   
R1). $h(5 L
a( ƒ .  U58

−7

10

3.5

4

4.5

5 9

x 10

»º ¤iyfº?²h¤Cz

SeÏ

"J - i"'e
( <  e(
- 9!Q a ">u e (
U5{e58
M  !
[
m,n.9!/ V(52 "B5n# 
c(
9((53
U GÕh$9

Ï

Ý Ô

§i¨òÎ

Ý

§]¨

Ô

£ïBÏ Î

ï 1
 (
möy(5: "B
',?$
9)%,  Mbc*+, .-N% /'!G } "au/'
 !
I/0!
\ä ü ¦vŒ qî_†Š J36GyY
ï Ý

°

.¯


' Ï

(5+kPRA^Tê c1) "
9X(5^ YJ7 a(;1)
m1!1
 f

Õh 9!=… 4J2)/,  B ")(%(,  )-!, )(57(
)(53 "!1/0

§]«

’S
!. "3kP&A^T•1,!/-!E / ? Y ((
V "i(
%0e1+$,?=U
" d=^(
X ',i?? G



 )5f "[†mîq†@1).  u@-((, y (
U5{a5{
  !
w.-  } "BU . @/'(; 1
 !


"V58#
M=(
9!B(5&
 MU G

6­

öm¦Lˆ(ŠJ2(+  
^5{(ekPRASUTv 1). M"N "eu( S (
U 5{M35{
  !
]G }  "  "'.

 1
 !
i!"J2- ! &
 R1) "B " u( R (
U5{R58
M  !
N-!)J2, ,]
.^ "e" 9!"\58# 
 ‘VÕh 9!VG

 , , U . ; ">" 9!"Mf((c52 ";kPRASUTW@( 1, 7-G4 "ePRASUTNHL71, G

6.1 PEEC Circuit } "=€A U ;u(1/, a$)c? ,? 
9m5n!1

6.3 A Mechanical System t4u/, !? 
9e "7#  -((, 
Vj!!G 9G 6„?†ˆ‡{ƒR%0 YJ 
ASChDy?  h(
'V1"'
 (, UU 1%0 "BPRA^TNHy(
>kP&A^T•(
"=(@, ,$

P&t&t Dp   d. $(
æ„ …(…(‡>5a(
o,  (1)(9(
  y/(%, 1NG } "3?? & &(
>ASChD[
 YJ (bV!
 U $
9e(5]… †Š(Š+(/(? (

(, S%02(//,  e ^@(h1, $
9S51M"(
 , UU 1Gfk"cUU 1+ >58(eMu1/, f$
c "f1,$
9=
'  1>,n.  !
c(5 TNt:TXkQ-$G^Y
cÕh 9!BŽ0J2f"J•!1;



†‹!…)$
 (](‰!‰!Š\
'  - f!/, 
9!i
'c\
9!, B U  - ^!S " 2 -  "S ? G }  " ^?? 3  58(1B,$ 

/'. !
)(5iPRASUTNHw(
')kP&A^T58(7/(/0  R(5]kP&A^T˜

7. CONCLUDING REMARKS ‚[>"'- B b 
ma58"O,  !bm V*+, .-X%/(M%(Q/

0

10

Exact PRIMA model SPRIM model

”U  !
a M"
$# M"a(7PRASUTNHf 58(3@(31 , 
9B(5i,n.9!eASChDy?? GR‚[^"'- S"J7
\ " $
\(2  !% (
QP&A^TNH7@,$ b V1(1
1). M"
9f/!/0M Y '(
f 

−1

10


 V
c \/.”UM B(
N)%'( V1) ?uX58(e "B
,?
9 %,  bc*+, .-c%/'(!h% e(

O)
c1) ?ucJ7"!

V1int/V1ext

(,$1
y/'(
y(
(
> " >%,  MbO*+, .-[%/'!Gy~ ( !
` "$f
 9!" RJ2\/(/0!yN
-!,2/.”U  !
` "
$# (

−2

10

"kPRASUT

(, 9! "1NE " V/- f(, ,R ";M$(,&U M

'"XS/( -   / ? Y '(
=(
'@75{1N (5^ASChDo?? GQY
O/'  ,$&
) "akPRA^T



(71,$2%$9! 
$-!? YyP42sSJ•Ü:(b0&s,n.  !
m(
'X " ,$  !
]GY


L2½ÂÀDJÖ?×j×^ëN¿ÄÈM½¿0¾ÄjÅÂ(¿0¾(É3Î&Â(¿ÌBÈM½ÈM¿'ÀMÈ\Â(¿%ˆÅ =BÆÉ ¾(ÄjÅÂ(¿yÂB ˆ ÈsB 6 = ÅÀÂ(¿8 ÆÀÄ@½2L ½ÂÀÈMÃÃÈMÃ=¾(¿d B Ý ÈMÇÅjÀMÈMÃ&/'9!\†!†'Ï †(† PR$ J3 'sSJÒ! Í  ]†‰(‰ ‹ G0Ut&t:t3G

„?†‡VAVG ‚gGÕ
'
')PhGÕ,$1)

iG!AS@(21,? 
954,$9!f, 
.,?   1
(,6?? 7
9

‡2L ½ÂÀDf Ý ÈMÃÅò(Š ¿HGŸ^P ÆÄ@Âf = ¾? d UE0ÈMÃÄ
(

 ;1)!1, 
9Q(,?

R0E ÈÀÁ:D³+Ý ÅAŠ©DÖ?×j×^ë¨NՉÕÄÕ*á6P¨Î³OôN ¿ÄÈM½M¿'¾(ÄjÅÂ(¿'¾!É Î&Â(¿ÌBÈM½ÈM¿'ÀMÈ[¿äÎ&Âf=SÊ'ÆÄÈM½ P^ÅS

È %ÝfÈMÃÅAŠ(¿S/'9!cŒˆÏ  Œ

9!? "1NGY


Ci(fHS,$(1? !4D2(, ?5{
$3†‰!‰ ‹ GhYt:t&tgD (1/ MBk ?  =P4G „ … †‡fC&G }

G]P& , ,$(9!;(
'NAVG6HVG'A^!"MG6HSU1/   VJ2- 5{1

HNÕÄÕÄÕ\E½U¾(¿ÃDÄÎ&Â.=(
'aC&G }

„?†Œ.‡VAVG4‚gG:Õ
y(
'IPhG&Õ,$1)

]G }  " Xk TNP&l