Embed Size (px)
Citation preview
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 interconnections
Z = 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
dadlAa
1
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
jr
dadl
a
IA
4
jconductorofareacrossa
jconductorofelementdl
rrrwhere
j
j
jiij
:
:
,
i a j a
jiij
ji
jiij
i j
dadar
dldl
aaL
1
4
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
b
mkkm
mk
mkij
k m
k
k
m
m
dadar
dldl
aaL
1 1
1
4
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
dldl
aaL
1
4
Loop inductance between loop i and j is
Partial Inductance (cont’d)
k m
k
k
m
m
km
a a
c
b
c
b
mkkm
mk
mkP dada
r
dldl
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
TW
l
l
Le
i
ii log2
12ln002.0
l
TW
TW
l
l
L
i
ii )(2235.0
2
12ln002.0
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
33360
1ln
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
zyxzz
yxyx
xz
zyxyy
zxxz
zy
zyxxx
zyzy
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
1ln
242060
11
60
1
20ln
24
1
tan6
1
4tan
64tan
6420
1
60ln
2460ln
24
11ln
24
2
AuAAu
Au
Au
Au
AuAA
uAA
uA
uAA
uAAu
A
uA
A
uA
u
uA
uA
uAA
u
Au
AAu
AAu
AAu
AA
ul
L
i
ii
2
47
1
46
3
45
224
223
22
21
lnln1
ln1
11
A
AuA
A
AA
A
AAuA
uAAuAW
T
W
lu
.1
111
ln31
1ln36
2
3
3
2
3
4
222
uu
uuu
uuuu
l
L
i
ii
where
If T/W < 0.01
Self Inductance (cont’d)
FastHenry’s Formula
aratarawarraratatawawar
atawar
arawawwwrrar
wawwrar
aratatttrrar
tattrar
arw
t
t
w
art
w
w
t
ar
tw
tw
arataw
wt
wtarawat
tw
twarawrw
t
t
w
aratrt
w
w
t
arraww
t
t
w
arratt
w
w
t
raw
t
tat
w
wLii
111
20
1
))(1)(1)((
)1(
))()()((
)(
))()()((
)(
60
1tantantan
1
6
1
)(sinh
1
)(sinh
1
)(sinh
)(sinh
)(sinh
)(sinh
24
11sinhsinh
1sinh
1
4
12
2
2111
21
2
21
2
21
2
221
2
2
12
12
111
111 2222
22
twartatwaw
twrl
Tt
l
Wwwhere
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 rgrgggl
L
22222222
1111 sinhsinhsinhsinh001.0
dddd
ddddLij
where pgvpgvpgpgl
DyDxr
l
Dzp
l
lv
iii
j
4321
22
11
mlmlxxx 1lnsinh 21where
Mutual Inductance (cont’d)
Hoer’s Formula (multiple filaments)
)()()(tan6
tan6
tan6
33360
1ln
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
zyxzz
yxyx
xz
zyxyy
zxxz
zy
zyxxx
zyzy
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
