OutLine Introduction – On-Chip Inductance Loop Inductance and Partial Inductance Closed Forms of Inductance Formulae
Self Inductance Formulae- Hoer, FastHenry, Ruehli, Grover
Mutual Inductance Formulae- Hoer (FastHenry), Ruehli, Grover
Computational Results Conclusion
Introduction – On-Chip Inductance As the clock frequency grows fast, the reactance becomes
larger for on-chip interconnectionsZ = R + jwL w is determined not by clock frequency itself but by clock
edge
w ~ 1/(rising time)
More layers are applied, wider conductors are used Wide conductor => low resistance
Multiple layer interconnections make complex return loops Inductance is defined in the closed loop in EM
Loop Inductance
jkifIforI
L kj
ijij 0
i a
iiiji
ij
i
dadlAa1
Loop inductance is defined as the induced magnetic flux in the loop by the unit current in other loop
where, represents the magnetic flux
in loop i due to a current Ij in loop j
ijLoop i Loop j
Ij
ij
The average magnetic flux can be calculated by magnetic vector potential Aij
ij
where, ai represents a cross section of loop i
Loop Inductance (cont’d)
j a ij
jj
j
jij
jrdadl
aI
A4
jconductorofareacrossa
jconductorofelementdl
rrrwhere
j
j
jiij
:
:
,
i a j a
jiij
ji
jiij
i j
dadar
dldlaa
L 14
The magnetic vector potential A, defined by B = A, has an integral form
So, loop inductance is
Partial Inductance Problems of loop inductance
The loops (called return paths) are hardly defined explicitly in VLSI
In most cases, the return paths are multiple
Partial inductance proposed by A. Ruehli The return path is assumed at infinite
for each conductor segment It can be directly appliable to circuit simulator like SPICE
1
2
3
4
5
Partial Inductance (cont’d)
K
k
c
bi
k
k1
K
k
M
m a a
c
b
c
bmk
km
mk
mkij
k m
k
k
m
m
dadar
dldlaa
L1 1
14
M
m
c
bj
m
m1
(assume loop i consists of K segments and loop j does M segments)So, loop inductance is
i a j a
jiij
ji
jiij
i j
dadar
dldlaa
L 14
Loop inductance between loop i and j is
Partial Inductance (cont’d)
k m
k
k
m
m
km
a a
c
b
c
bmk
km
mk
mkP dada
rdldl
aaL ||1
4
K
k
M
mPkmij km
LSL1 1
Definition of partial inductance
The sign of partial inductance is not considered So, partial inductance is solely dependent of conductor geometry
Sign rule for partial inductance
where, Skm = +1 or –1
The sign depends on the direction of current flow in the conductors
Geometry and Formulae Conductor Geometry
Inductance Formulae Self Inductance : Grover(1962), Hoer(1965), Ruehli(1972),
FastHenry(1994) Mutual Inductance : Grover(1962), Hoer(FastHenry)(1965), Ruehli(1972)
x z
Dx
Dy
Dz
y
Conductor 1 Conductor 2
T
W
l
(a) Single Conductor (b) Two Parallel Conductors
Self Inductance Grover’s Formula
e
TWl
lL
ei
ii log212ln002.0
l
TWTW
llL
i
ii )(2235.0212ln002.0
T/W logee T/W logee T/W logee T/W logee
0 0 0.2 0.00249 0.5 0.00211 0.8 0.00181
0.05 0.00146 0.3 0.00244 0.6 0.00197 0.9 0.00178
0.1 0.00210 0.4 0.00228 0.7 0.00187 1.0 0.00177
Grover 2(without table)
Self Inductance (cont’d)
Hoer’s Formula
)()()(tan6
tan6
tan6
333601ln
24244
ln24244
ln24244
008.0
000
222
13
222
13
222
13
222222222444
22
2224422
22
2224422
22
2224422
22
zyxzyxx
yzyzx
zyxy
xzzxy
zyxz
xyxyz
zyxxzzyyxzyxyx
zyxzzyxyx
xz
zyxyyzxxz
zy
zyxxxzyzy
TWL
lTW
ii
2
1
2
1
2
1
1 ),,()1()()()(),,(1
2
1
2
1
2i j k
kjikji
s
s
r
r
qq srqfzyxzyxfwhere
Self Inductance (cont’d)
Ruehli’s Formula
3142
3
63
3
51
2
3
4314222412712
4
17
4
16
4
12
542
2
2
73
2
34
2
6252
2
60ln
24
1ln242060
11601
20ln
241
tan61
4tan
64tan
64201
60ln
2460ln
2411ln
242
AuAAuAu
Au
Au
AuAAuAAu
Au
AAuAAu
AuA
AuAu
uAuAuAA
u
Au
AAuAAu
AAu
AAul
L
i
ii
2
47
1
46
3
45
224
223
22
21
lnln1ln1
11
AAuA
AAA
AAAuA
uAAuAWT
Wlu
.1111ln311ln36
23
32
34
222
uu
uuu
uuuu
lL
i
ii
where
If T/W < 0.01
Self Inductance (cont’d)
FastHenry’s Formula
aratarawarraratatawawaratawar
arawawwwrrarwawwrar
aratatttrrartattrar
arwt
tw
artw
wt
artw
tw
aratawwt
wtarawattw
twarawrwt
tw
aratrtw
wt
arrawwt
tw
arrattw
wt
rawt
tatw
wLii
111201
))(1)(1)(()1(
))()()(()(
))()()(()(
601tantantan1
61
)(sinh1
)(sinh1
)(sinh
)(sinh
)(sinh
)(sinh
2411sinhsinh1sinh1
412
2
2111
21
2
21
2
21
2
221
2
2
12
12
111
111 2222
22
twartatwaw
twrlTt
lWwwhere
Comparisons of Self Inductance
Formula Short Conductor(l/W < 10)
Medium Conductor
(10 < l/W < 1000)
Long Conductor(l/W > 1000)
Hoer O O X
FastHenry O O O
Ruehli X O X
Ruehli (T=0) O (30% larger)(T/W < 0.01)
O(T/W < 0.01)
O(T/W < 0.01)
Grover X O O
Grover2 X O O
Mutual Inductance Ruehli’s Formula
Grover’s Formula (single filament)
4
1
22221 ln14 m
mmmmm
i
ij rgrggglL
22222222
1111 sinhsinhsinhsinh001.0
dddd
ddddLij
where pgvpgvpgpgl
DyDxr
lDzp
ll
viii
j
4321
22
11
mlmlxxx 1lnsinh 21where
Mutual Inductance (cont’d)
Hoer’s Formula (multiple filaments)
)()()(tan6
tan6
tan6
333601ln
24244
ln24244
ln24244
001.0
21
12
21
12
21
12
,
,
,
,
,
,
222
13
222
13
222
13
222222222444
22
2224422
22
2224422
22
2224422
2121
zyxzyxx
yzyzx
zyxy
xzzxy
zyxz
xyxyz
zyxxzzyyxzyxyx
zyxzzyxyx
xz
zyxyyzxxz
zy
zyxxxzyzy
TTWWL
lDlD
DllD
TDTD
DTTD
WDWD
DWWD
ij
zz
zz
xx
xx
yy
yy
where
4
1
4
1
4
1
1,
,
,
,
,, ),,()1()()()(),,(
31
42
31
42
31
42i j k
kjikji
ss
ss
rr
rr
qqqq srqfzyxzyxf
Conclusion On-Chip inductance becomes a troublemaker in high-
performance VLSI design Higher clock frequency, wide interconnections, complex
return paths The concept of partial inductance is useful in VLSI area
Not related to the return path Only dependent of geometry
Several inductance formulae are in hand but they have Different computational complexities Different applicable ranges according to the geometry