Chapter 4 Interconnect Analysis

Organization

• 4.1 Linear System• 4.2 Elmore Delay• 4.3 Moment Matching and Model Order Reduction

– AWE

– PRIMA

• 4.4 Recent development– MOR for network

– Parameterized MOR

Reading Assignment for 4.1 and 4.2

• Elmore delay model (Elmore, Journal of Applied Physics, 1948 – http://eda.ee.ucla.edu/EE201A-04Spring/elmore.pdf

• Elmore delay for RC tree (Rubinsteun-Penfield-Horowitz,TCAD'83– http://eda.ee.ucla.edu/EE201A-04Spring/Elmore_TCAD.pdf

Chapter 4.1 Linear System

• Laplace Transformation

• Pole/residue

• Basic Circuit Analysis

Laplace Transformation

• Definition:dtetfsFtftf st

0

)()()}({ )( Ltime domain frequency domain

Time domain (t domain)

Complex frequency domain (s domain)

Linear Circuit

Differentialequation

Responsewaveform

Laplace Transform

Inverse Transform

Linearequation

Responsetransform

L

L-1

Frequency Domain Transfer Functionand Time Domain Impulse Response

Frequency domain representationFrequency domain representation

H(s)u(s) y(s) = H(s) u(s)

Linear systemLinear system

h(t)u(t)

duthty )()()(

Linear systemLinear system

Time domain representationTime domain representation

The transfer function H(s) is the Laplace Transform The transfer function H(s) is the Laplace Transform of the impulse response h(t)of the impulse response h(t)

Circuit Analysis Using Laplace Transforms

Time domain (t domain)

Complex frequency domain (s domain)

Linear Circuit

Differentialequation

Classicaltechniques

Responsewaveform

Laplace Transform

Inverse Transform

Algebraicequation

Algebraictechniques

Responsetransform

L

L-1

Poles and Zeros of F(s)

• Scale factor: K = bm/an

• Poles: s = pk (k = 1, 2, ..., n)

• Zeros: s = zk (k = 1, 2, ..., m)

01

1

1

01

1

1)(asasasa

bsbsbsbsH

n

n

n

n

m

m

m

m

)())((

)())(()(

21

21

n

m

pspsps

zszszsKsH

Resonant frequencies

Pole-Zero Diagrams

1 :pole1

1)(

ss

sF

j

1

s-plane

pole locationzero location

j

j

j

s-plane

js

s

s

sAsF

:poles

:zero

)(

)()(

22

j

s-plane

00:pole

1)(

jss

sF

Poles and WaveformsIf poles in right-plane,

waveform increases withoutbound as time approaches infinity

If poles on j-axis,waveform neither decays nor grows

If poles in left-plane,waveform decays to zero

as time approaches infinity

Real poles produce exponential waveforms

Complex poles come in pairs that produce

oscillatory waveforms

Basic Circuit Analysis

• Output response

• Basic waveforms– Step input

– Pulse input

– Impulse Input

• Use simple input waveforms to understand the impact of network design

Network structures & state

Input waveform & zero-states

Natural response vN(t)(zero-input response)

Forced response vF(t)(zero-state response)

For linear circuits: )()()( tvtvtv FN

unit step function

u(t)=0

1

0t

0t

1

pulse function of width T

0

1/T

-T/2 T/2

)

2()

2(

1)(

Ttu

Ttu

TtPT

unit impulse function

1)(

0any for s.t.

0for singular

0for 0)(

0when )( : )(

dtt

t

tt

TtPt T

dt

tduδ(t)

dxxtut

)(or

)()(

definitionBy

Inputs

Time Moments of Impulse Response h(t)

• Definition of moments

)()( sHth L

dttthsi

dtsti

thdtethsH

i

i

i

i

ist

00

00

0

)()(!

1

)(!

1)()()(

i-th moment dttthi

m iii

0)()1(

!

1

Chapter 4.2 Elmore Delay

• Lumped and distributed interconnect delay model

• Elmore delay and distributed interconnect delay model

• Elmore delay and time moments

Interconnect ModelLumped vs Distributed

Lumped Distributed

R

C

r

c

r

c

r

c

r

c

Analysis of Simple RC Circuit

0)()(

tvdt

tdvRCzero-input response:

(natural response)

step-input response:

match initial state:

output responsefor step-input:

v0v0u(t)

v0(1-eRC/T)u(t)

RCt

N Ke(t)vRCdt

dv(t)

v(t)

11

)()()(

0 tuvtvdt

tdvRC

)()()()( 00 tuvKetvtuvtv RCt

F

)()1()( 0 tuevtv RCt

0)( 0)0( 0 tuvKv

RC-Tree

– The network has a single input node– All capacitors between node and ground– The network does not contain any resistive loop

R1

C1

s

R 2

C2

R 4

C4

C3

R3

Ci

Ri

1

2

3

4

i

RC-tree Property

– Unique resistive path between the source node s and any other node i of the network path resistance Rii

Example: R44=R1+R3+R4

R1

C1

s

R 2

C2

R 4

C4

C3

R3

Ci

Ri

1

2

3

4

i

RC-tree Property

– Extended to shared path resistance Rik:

Example:Ri4=R1+R3

Ri2=R1

)])()([( s.t. kspathispathRRR jjik

R1

C1

s

R 2

C2

R 4

C4

C3

R3

Ci

Ri

1

2

3

4

i

Elmore Delay

• Assuming:– Each node is initially discharged to ground

– A step input is applied at time t=0 at node s

• The Elmore delay at node i is:

• It is an approximation: it is equivalent to first-order time constant of the network– Proven acceptable

– Powerful mechanism for a quick estimate

N

kikkDi RC

1

RC-chain (or ladder)

• Special case

• Shared-path resistance path resistance

N

kkkkDN RC

1

R1

C1

R2

C2

RN

CN

Vin VN

RC-Line Delay

22

1

2

)1( 22

11

rcL

N

NrcL

NNRCCkRRC

N

N

k

N

kkkkDN

R

C

R

C

R

C

Vin VN

R=r · L/N

C=c·L/N

– Delay of wire is quadratic function of its length– Delay of distributed rc-line is half of lumped RC

Time Moments of Impulse Response h(t)

• Definition of moments

)()( sHth L

dttthsi

dtsti

thdtethsH

i

i

i

i

ist

00

00

0

)()(!

1

)(!

1)()()(

i-th moment dttthi

m iii

0)()1(

!

1

• Note that m1 = Elmore delay when h(t) is monotone voltage response of impulse input

Elmore Delay for RC Trees

• Definition– h(t) = impulse response

– TD = mean of h(t)

= • Interpretation

– H(t) = output response (step process)– h(t) = rate of change of H(t)

– T50%= median of h(t)

– Elmore delay approximates the median of h(t) by the mean of h(t)

0

dtt h(t) medianof v’(t)(T50%)

v'(t)

dtttvTD

ofmean

)('0

h(t) = impulse response

H(t) = step response

Elmore Delay in RC Tree

kkCkiRiDTi

kjkPjP

jkR

iis

iiP

:is node delay to Elmore :Theorem

& input to from

pathcommon of resistance:

at rooted subtree:

; node input to frompath :

input

i

k

jSi

path resistance Rii

Rjk

Proof of Theorem

0 ))(1()1)((lim

]0 )()([lim

0 )(0|)(0 )('

Therefore

)()(

) and between res.path (common ) cap o(current t

)in scap' all current to(

on drop voltageThe)(1

)(

i node of cap. current toLet

dtti

vTTi

vT

dtT ti

vTTi

vT

dtti

vtti

vdttti

vi

DT

k dt

tk

dv

kC

kiR

kiR

k dt

tk

dv

kC

k kP

iPk

iPk i

Sk

Ri

Pti

vdt

ti

dvCi