ECE 546 – Jose Schutt-Aine 1
ECE 546Lecture -13
Latency Insertion MethodSpring 2014
Jose E. Schutt-AineElectrical & Computer Engineering
University of [email protected]
ECE 546 – Jose Schutt-Aine 2
- Up to 16 layers- Hundreds of vias- Thousands of TLs- High density- Nonuniformity
Vertical parallel-plate capacitance 0.05 fF/mm2
Vertical parallel-plate capacitance (min width) 0.03 fF/mmVertical fringing capacitance (each side) 0.01 fF/mmHorizontal coupling capacitance (each side) 0.03
Mitsumasa Koyanagi," High-Density Through Silicon Vias for 3-D LSIs"Proceedings of the IEEE, Vol. 97, No. 1, January 2009
TSV Density: 10/cm2 - 108/cm2
Source: M. Bohr and Y. El-Mansy - IEEE TED Vol. 4, March 1998
Packaging Complexity
Chip Complexity
Challenges in Integration
ECE 546 – Jose Schutt-Aine 3
X ce lls
Y ce lls
.+- +-
+
-
+- +-
+
-
+
-
=Unit cell
Typical workstation simulation time for a 1200-cell network is 2 h 40 min.
Too time consuming!
- Determine R,L,G,C parameters and define cell- Synthesize 2-D circuit model for ground plane- Use SPICE (MNA) to simulate transient
Modeling
PDN Modeling
ECE 546 – Jose Schutt-Aine 4
• MNA has super-linear numerical complexity• LIM has linear numerical complexity• LIM has no matrix ill-conditioning problems• Accuracy and stability in LIM are easily controlled• LIM is much faster than MNA for large circuits
Why LIM?
ECE 546 – Jose Schutt-Aine 5
.
+- +-
+
-
+
-
+
-
+- +-
+
-
+
-
+
-
+- +-+
-
+
Latency Insertion Method
ECE 546 – Jose Schutt-Aine 6
Each branch must have an inductor*
Each node must have a shunt capacitor*
Express branch current in terms of history of adjacent node voltages
Express node voltage in terms of history of adjacent branch currents
Latency Insertion Method
* If branch or node has no inductor or capacitor, insert one with very small value
ECE 546 – Jose Schutt-Aine 7
• Represents network as a grid of nodes and branches
• Discretizes Kirchhoff's current and voltage equations
• Uses ”leapfrog” scheme to solve for node voltages and branch currents• Presence of reactive elements is required to generate latency
1 1/2 1/2n n n n nij ij i j ij ij
ij
tI I V V R I
L
1/2
1/2 1
aNnn ni ii ik
n ki
ii
CVH I
tVC
Gt
Node structureBranch structure
LIM Algorithm
ECE 546 – Jose Schutt-Aine 8
1 1/2 1/2n n n n nij ij i j ij ij
ij
tI I V V R I
L
1/2
1/2 1
aNnn ni ii ik
n ki
ii
CVH I
tVC
Gt
1 2
1/2 3/2 5/2
n n nij ij ij
n n ni i i
I I I
V V V
n: time
LIM: Leapfrog Method
Leapfrog method achieves second-order accuracy, i.e., error is proportional to Dt2
ECE 546 – Jose Schutt-Aine 9
1 1/2 1/2 1/2n n n n n nij ij i j ij ij ij
ij
tI I V V R I E
L
For branch =1, Nb
Update current as per Equation:
For time=1, Nt
Next branch;For node=1, Nn
Update voltage as per Equation
1/2
1 11/2
iMn Nn n ni ii ik q
k qni
ii
CVH I J
tV
CG
t
Next node;Next time;
timeloop
branchloop
nodeloop
LIM Code
ECE 546 – Jose Schutt-Aine 10
No of Nodes
20,000 30,000 40,000 50,000
SPICE(sec)
1224 2935 4741 7358
LIM(sec)
9 13 17 21
Speedup 136 225 278 350
Simulation Times
Standalone LIM vs SPICE
ECE 546 – Jose Schutt-Aine 11
0 0.5 1 1.5 20
20
40
60
80
100
120
140
Log Size (normalized to 2,000 nodes)
Spe
edup
Fac
tor
LIM v.s. SPICE
LIM vs SPICE
ECE 546 – Jose Schutt-Aine 12
Mutual inductanceuse reluctance (1/L) Branch capacitors special formulation Shunt inductors special formulation Dependent sourcesaccount for dependenciesFrequency dependence use macromodels
LIM: Problem Elements
ECE 546 – Jose Schutt-Aine 13
We wish to determine the current update equations for that branch.
In order to handle the branch capacitor with LIM, we introduce a series inductor L.
LIM Branch capacitor
ECE 546 – Jose Schutt-Aine 14
1/2 1/2n nn c cij
V VI C
t
1/2ncV
1/2 1/2n n nc c ij
tV V I
C
In the augmented circuit, we have:
where Vc is the voltage drop across the capacitor; Vc=Vo-Vj. Solving for
LIM Branch capacitor
ECE 546 – Jose Schutt-Aine 15
11/2 1/2 1/2 1/2
n nij ijn n n n
L i j c
I IV V V V L
t
1 1/2 1/2 1/2n n n n nij ij i j c
tI I V V V
L
so that
which leads to
LIM Branch capacitor
11/2
n nij ijn
L
I IV L
t
The voltage drop across the inductor is given by:
ECE 546 – Jose Schutt-Aine 16
2t
C
In order to minimize the effect of L while maintaining stability, we see that
- L must be very small
- L must be A large branch capacitor helps as well as
small time steps.
LIM Branch capacitor
2
1 1/2 1/2 1/21n n n n nij ij i j c
t tI I V V V
LC L
After substitution and rearrangement, we get:
ECE 546 – Jose Schutt-Aine 17
C LpVi Vj
Iij
Apply the difference
equation directly to branch
1/ 2
0
11/ 2 1 1/ 2
1
knijn n
c ijk
cn nijij ijn n n n n
L p ij ij ij ijij p p
L p nijn n
ij ij
I CV T
tdV
I CI I t tdt
V L I I V TdI t L L
V Ldt I C
T Tt
LIM Branch capacitor
ECE 546 – Jose Schutt-Aine 18
21 2
1 1 2
1
in / n / Mn ni i
i i ijn /
ikk
i
V VC G IV H
t
21 2
1 1 2
1
in / n / Mn ni i
i i ijn /
ikk
i
V VC G IV H
t
1 2 1 2
1
1 2 1 2
2
in / n /
i i
n / n / Mn ni i i
i ij ikk
V VI
tVC V
GH
LIM Node Formulations
Explicit
Implicit
Semi-Implicit
ECE 546 – Jose Schutt-Aine 19
LIM Branch Formulations
Explicit
Implicit
Semi-Implicit
11 2 1 2 1 2
n nij ijn n
ij/ n / n /
i j ij ij ij
I IV V L R EI
t
11 112 1 2 2
n nij ijn / n / n /
i jnijij ij ij
I IV V L R E
tI
1
1 2 1 2 11 2
2
n nij ij ij n n
ij ijn / n / n /
i j ij ij
I I RV V I IL E
t
ECE 546 – Jose Schutt-Aine 20
1 1 2 1 2 1 2/ / /n n n n n nij ij i j ij ij ij
ij
tI I V V R I E
L
1 2
1 2 1
/
/
iMnn ni ii ik
n ki
ii
CVH I
tV
CG
t
Ey
Ey
Ey
Ey
HyHy
Ex
Ex
ExEx
Hx
Hx
Ez
Ez
Ez
Hz
Hz
x
y
z
1 1 2 1 2
1 2 1 2
1
1
/ /
/ /
( , , ) ( , , ) ( , , )
( , , ) ( , , )
n n n nx x z z
n ny y
c tE i j k E H i j k H i j k
y
c tH i j k H i j k
z
1 2 1 2 1
1
/ /( , , ) ( , , ) ( , , )
( , , ) ( , , )
n n n nx x z z
n ny y
c tH i j k H E i j k E i j k
y
c tH i j k E i j k
z
Latency Insertion Method (LIM)Yee Algorithm
Field Solution Circuit Solution
FDTD & LIM
Lij
VjVi
Rij
Iij
Eij
Ci GiHi
t LC
Courant-Friedrichs-Lewy criteria stability condition
ECE 546 – Jose Schutt-Aine 21
0 10 20 30 40 50 60 70 80 90 100
0
1
2
3
4
5
ns
Vol
t
v1
v2v3
max0.99 t
-2 0 2 4 6 8 10 12 14 16 18 20
0
1
2
3
4
5
ns
Vol
t
v1
v2v3
max1.01 t
max = 0.20099505 nst
LIM: Stability Analysis
ECE 546 – Jose Schutt-Aine 22
1 2 1 21 2 1 2
1 2 1 2
1
2
2
i
n / n /n / n / n ni i i
i i i i Li
Mn / n / n nik
k k ip p ijj
V V GC V V H I
t
BV V S I I
LIM and Dependent Sources
ECE 546 – Jose Schutt-Aine 23
1
2
1
2
t
n+1/2 n-1/2n+1/2 n 1/2 n n
L
n+1/2 n 1/2 n n
v - vC G v v h i
B v v Si = -Mi
LIM and Dependent Sources
ECE 546 – Jose Schutt-Aine 24
where M is the incidence matrix. M is defined as follows
1qpM if branch p is incident at node q and the
current flows away from node q.
1qpM if branch p is incident at node q and the
current flows int o node q.
0qpM if branch p is not incident at node q.
LIM and Dependent Sources
ECE 546 – Jose Schutt-Aine 25
1 1 1 1 0 0 0
2 1 0 0 1 0 1
3 0 0 1 1 1 0
4 0 1 0 0 1 1
branch
a b c d e f
node
node
node
node
M
Incidence Matrix
ECE 546 – Jose Schutt-Aine 26
tn+1 n -1 n+1/2L L ni = i L v
1
2t
n+1/2 n-1/2' n+1/2 n 1/2 n n ' n
L
v - vC G v v h i = -M i
'G = G + B 'M = M + S
1
2 2t t
' 'n+1/2 n 1/2 n n ' n
L
C G C Gv v h i M i
LIM and Dependent Sources
ECE 546 – Jose Schutt-Aine 27
1
1
2
2
t
T n+1/2 n+1 n n n n+1/2 n+1/2c
n+1/2 n n
L RM v i - i i i e v
ZTv i i
11 2 1 2 1 1 2 1 2
1 2 1
2
2
n nij ij ijn / n / n n n / n /
i j ij ij ij ij cij
ijpqn / n nijk k pq pq
I I RV V L I I E V
t
ZT V I I
LIM and Dependent Sources
ECE 546 – Jose Schutt-Aine 28
t n+1/2 n-1/2 -1 nc c bv v C i
tn+1 n -1 n+1/2L L ni = i L v
1
2 2'
t t
' 'n+1 n n+1/2 n+1/2 T n+1/2
c
L R L Ri i e v M v
1
2 2t t
' 'n+1/2 n 1/2 n n ' n
L
C G C Gv v h i M i
LIM and Dependent Sources
ECE 546 – Jose Schutt-Aine 29
1 1
1
2 2 2
2
t t t
t
n+1 n T' n+1/2
n+1/2c
L R' L R' L R'i i M v
L R'v
1 1
1
2 2 2
2
t t t
t
n+1/2 n 1/2 n
nL
C G' C G' C G'v v M'i
C G'i
LIM and Dependent Sources
ECE 546 – Jose Schutt-Aine 30
1
2t
+
C G'P 2t
-
C G'P
1
2t
L R'Q
2t
L R'Q
n+1 n T' n+1/2+ - +i = Q Q i + Q M v
n+1/2 n 1/2 n+ - +v P P v P M'i
n+1 n T' n-1/2 T' n+ - + + - + +i = Q Q i + Q M P P v - Q M P M'i
DEFINE:
so that
Stability Analysis
ECE 546 – Jose Schutt-Aine 31
2 1 2/ /
n+1 n+ - +T' T'n+1 n
+ + - + - + +
P P -P M'v v
Q M P P Q Q - Q M P M'i i
n+1 T' n-1/2 T' n
+ + - + - + +i = Q M P P v + Q Q - Q M P M' i
n+1/2 n-1/2 n+ - +v = P P v - P M'i
The 2 matrix equations can be combined in a single matrix equation that reads
2 1 2/ /
n+1 n
n+1 n
v vA
i ior
Stability Analysis
ECE 546 – Jose Schutt-Aine 32
+ - +T' T'
+ + - + - + +
P P -P M'A
Q M P P Q Q - Q M P M'
11 + -A P P 12 +A = -P M'
T'21 + + -A = Q M P P
T'22 + - + +A = Q Q - Q M P M'
A is the amplification matrix. All the eigenvalues of A must be less than 1 in order to guarantee stability. Therefore to insure stability, we must choose t such that all the eigenvalues of A are less than 1
Amplification Matrix
ECE 546 – Jose Schutt-Aine 33
Stability Methods for LIM
• LIM is conditionally stable upper bound on the time step Δt.
• For uniform 1-D LC circuits [2]:
• For RLC/GLC circuits [3]:
• For general circuits [4]:
[2] Z. Deng and J. E. Schutt-Ainé, IEEE EPEPS, Oct. 2004.[3] S. N. Lalgudi and M. Swaminathan, IEEE TCS II, vol. 55, no. 9, Sep. 2008.[4] J. E. Schutt-Ainé, IEEE EDAPS, Dec. 2008.
t LC
,1 12min min( )
in bN N
ii pii p
b
Ct L
N
Amplification matrix.
+ - +T T
+ + - + - + +
P P P M
Q M P P Q Q Q M P MA
ECE 546 – Jose Schutt-Aine 34
Example – RLGC Grid
* SPECTRE – A SPICE-like simulator from Cadence Design Systems Inc.* Current excitation with amplitude of 6A, rise
and fall times of 10 ps and pulse width of 100 ps.
ECE 546 – Jose Schutt-Aine 35
Example – RLGC Grid
• Comparison of runtime for LIM and SPECTRE*.
• LIM exhibits linear numerical complexity!– Outperforms conventional SPICE-like simulators.
• Expand on LIM as a multi-purpose circuit simulator.
Circuit Size
# nodes(SPECTRE)
SPECTRE (s) LIM (s)
20 × 20 1160 0.15 1.53
40 × 40 4720 2.14 6.25
60 × 60 10680 25.73 14.23
80 × 80 19040 140.59 25.71
100 × 100 29800 445.77 40.64
120 × 120 42960 945.00 62.89
* Intel 3.16 GHz processor, 32 GB RAM.
ECE 546 – Jose Schutt-Aine 36
-0.2
0
0.2
0.4
0.6
0.8
1
Volts
0 5 10 15 20 25 30
Time (ns)
Drive Line at Near End
35 40
-0.15
-0.1
-0.05
0
0.05
0.1
0.15
0.2
Volts
0 5 10 15 20 25 30
Time (ns)
Sense Line at Near End
35 40
Cricket Software userdict /md known{/CricketAdjust true def}{/CricketAdjust false def}ifelse /mypsb /psb load def /mypse /pse load def/psb {} store /pse {} storecurrentpoint /picOriginY exch def /picOriginX exch defcurrentpoint pop /newWidth exch picOriginX sub defcurrentpoint /newHeight exch picOriginY sub def pop/newXScale newWidth 244 div def/newYScale newHeight 122 div def/psb /mypsb load store/pse /mypse load store
Computer simulations of distributed model (top) and experimental waveforms of coupled microstrip lines (bottom) for the transmission-line circuit shown at the near end (x=0) for line 1 (left) and line 2 (right). The pulse characteristics are magnitude = 4 V; width = 12 ns; rise and fall times = 1 ns. The photograph probe attenuation factors are 40 (left) and 10 (right).
LIM Simulations
ECE 546 – Jose Schutt-Aine 37
Twisted-pair cable Simulation setupGeneralized model [3]
Ideal line lossy linelossy line
Simulation results
Measured response
LIM Simulation Examples