1. Chapter 2 Interconnect Analysis
Prof. Lei He
Electrical Engineering Department
University of California, Los Angeles
URL: eda.ee.ucla.edu

2. Organization Chapter 2a Linear System
ECE902 VLSI Interconnect
Organization
Chapter 2a
Linear System
First Order Analysis (Elmore Delay)
Second Order Analysis
Chapter 2b Moment calculation and AWE
Chapter 2c Projection based model order reduction
Prepared by Lei He

3. Laplace Transformation
ECE902 VLSI Interconnect
Laplace Transformation
Definition:
time domain
frequency domain
Linear Circuit
Time domain
(t domain)
Complex frequency
domain (s domain)
Differential
equation
Linear
equation
Laplace Transform L
Inverse Transform
Response
waveform
Response
transform
L-1
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4. Poles and Zeros of H(s) Scale factor: K = bm/an
ECE902 VLSI Interconnect
Poles and Zeros of H(s)
Scale factor: K = bm/an
Poles: s = pk (k = 1, 2, ..., n)
Zeros: s = zk (k = 1, 2, ..., m)
Resonant frequencies
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5. Pole-Zero Diagrams pole location zero location s-plane s-plane s-plane
ECE902 VLSI Interconnect
Pole-Zero Diagrams
pole location
zero location
s-plane
s-plane
s-plane
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6. Poles and Waveforms If poles in right-plane,
ECE902 VLSI Interconnect
Poles and Waveforms
If poles in right-plane,
waveform increases without
bound as time approaches infinity
Complex poles come in pairs that produce oscillatory waveforms
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
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7. Basic Circuit Analysis
ECE902 VLSI Interconnect
Basic Circuit Analysis
Basic waveforms
Step input
Pulse input
Impulse Input
Use simple input waveforms to understand the impact of network design
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8. Inputs unit step function pulse function of width T
ECE902 VLSI Interconnect
Inputs
1/T 1
-T/2
T/2
unit step function
pulse function of width T
unit impulse function
u(t)= 1
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9. Time Moments of Impulse Response h(t)
ECE902 VLSI Interconnect
Time Moments of Impulse Response h(t)
Definition of moments
i-th moment
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10. Organization Linear System First Order Analysis (Elmore Delay)
ECE902 VLSI Interconnect
Organization
Linear System
First Order Analysis (Elmore Delay)
Second Order Analysis
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11. Interconnect Model Lumped vs Distributed
ECE902 VLSI Interconnect
Interconnect Model Lumped vs Distributed
Lumped
Distributed R C r c
How to analyze the delay for each model?
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12. Lumped RC Model R
v(t)
u(t) C
Impulse response and step response of a lumped RC circuit

13. Analysis of Lumped RC Model
ECE902 VLSI Interconnect R C
S-domain ckt equation (current equation)
Frequency domain response for step-input
Frequency domain response for impulse
match initial state: v0
v0(1-eRC/T)
Time domain response
for step-input:
Time domain response
for impulse:
Prepared by Lei He

14. Analysis of Lumped RC Model (cont’d)
Impulse response:

15. Delay for lumped RC model
V(t) 1v
Time Constant=RC
What is the time constant for more complex circuits?

16. Distributed RC-Tree The network has a single input node
ECE902 VLSI Interconnect
Distributed RC-Tree R1 C1 s R2 C2 R4 C4 C3 R3 Ci Ri 1 2 3 4 i
The network has a single input node
All capacitors between node and ground
The network does not contain any resistive loop
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17. ECE902 VLSI Interconnect
RC-tree Property R1 C1 s R2 C2 R4 C4 C3 R3 Ci Ri 1 2 3 4 i
Unique resistive path between the source node s and any other node i of the network  path resistance Rii
Example: R44=R1+R3+R4
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18. RC-tree Property Extended to shared path resistance Rik:
ECE902 VLSI Interconnect
RC-tree Property R1 C1 s R2 C2 R4 C4 C3 R3 Ci Ri 1 2 3 4 i
Extended to shared path resistance Rik:
Example: Ri4=R1+R3
Ri2=R1
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19. The Elmore delay at node i is:
ECE902 VLSI Interconnect
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:
Theorem: The Elmore delay is equivalent to the first-order time constant of the network
Proven acceptable approximation of the real delay
Powerful mechanism for a quick estimate
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20. ECE902 VLSI Interconnect
Proof of Theorem
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21. Interpretation of Elmore Delay
ECE902 VLSI Interconnect
Interpretation of Elmore Delay
median
of v’(t)
(T50%)
h(t) = impulse response
H(t) = step response
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)
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22. Elmore Delay Approximation
Elmore delay approximates 50% delay

23. RC-chain (or ladder) Special case
ECE902 VLSI Interconnect
RC-chain (or ladder)
Special case
Shared-path resistance path resistance R1 C1 R2 C2 RN CN
Vin VN
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24. RC-Chain Delay Delay of wire is quadratic function of its length
ECE902 VLSI Interconnect
RC-Chain Delay 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
Prepared by Lei He

25. Organization Linear System First Order Analysis (Elmore Delay)
ECE902 VLSI Interconnect
Organization
Linear System
First Order Analysis (Elmore Delay)
Second Order Analysis
Prepared by Lei He

26. Stable 2-Pole RC delay calculation (S2P)
The Elmore delay is the metric of choice for performance-driven design applications due to its simple, explicit form and ease with which sensitivity information can be calculated
However, for deep submicron technologies (DSM), the accuracy of the Elmore delay is insufficient

27. Moments of H(s)
Moments of H(s) are coefficients of the Taylor’s Expansion of H(s) about s=0

28. Driving Point Admittance
Let Y(s) be an driving point admittance function of a general RC circuit. Consider its representation in terms poles and residues
where q is the exact order of the circuit
Moments of Y(s) can be written as:

29. S2P Algorithm Compute m1, m2, m3 and m4 for Y(s)
Find the two poles at the driving point admittance as follows:
To match the voltage moments at the response nodes, choose
and the S2P approximation is then expressed as:
Note that m0* and m1* are the moments of H(s). m0* is the Elmore delay.

30. S2P Vs. Elmore Delay

31. Reading Assignment [1] Elmore delay model
[2] Elmore delay formula for RC tree
[3] S2P algorithm
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32. hw2
[1] Use Elmore model and two-pole model to approximate 50% delay for RC tree
[2] Moments for a linear network
[3] Expansion of projection based PRIMA