ECE 546 – Jose Schutt-Aine 1 ECE 546 Lecture -13 Latency Insertion Method Spring 2014 Jose E....

ECE 546 – Jose Schutt-Aine 1

ECE 546Lecture -13

Latency Insertion MethodSpring 2014

Jose E. Schutt-AineElectrical & Computer Engineering

University of [email protected]


- 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


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


• 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?


.

+- +-

+

-

+

-

+

-

+- +-

+

-

+

-

+

-

+- +-+

-

+

Latency Insertion Method


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


• 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


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


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


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


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


Mutual inductanceuse reluctance (1/L) Branch capacitors special formulation Shunt inductors special formulation Dependent sourcesaccount for dependenciesFrequency dependence use macromodels

LIM: Problem Elements


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


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



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


11/2

n nij ijn

L

I IV L

t

The voltage drop across the inductor is given by:


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.


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:


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



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


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


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


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


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


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



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.



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


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



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



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



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



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


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


+ - +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


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


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.


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.


-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


Twisted-pair cable Simulation setupGeneralized model [3]

Ideal line lossy linelossy line

Simulation results

Measured response

LIM Simulation Examples

ECE 546 – Jose Schutt-Aine 1 ECE 546 Lecture -13 Latency Insertion Method Spring 2014 Jose E....

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