1. הטכניון - מ.ט.ל. הפקולטה להנדסת חשמל - אביב תשס"ה
הטכניון - מ.ט.ל. הפקולטה להנדסת חשמל - אביב תשס"ה
4:36:48 PM
Modeling and Optimization of VLSI Interconnect Lecture 3: Interconnect modeling
Avinoam Kolodny
Konstantin Moiseev
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2. הטכניון - מ.ט.ל. הפקולטה להנדסת חשמל - אביב תשס"ה
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Outline
Recall of required material from signals & systems theory
LTI system, convolution, Laplace transform
Transfer function, impedance, admittance
Main signals: delta, step, ramp, saturated ramp and their responses
Distributed interconnect RC line
Derivation of heat transfer equation
Solution in s-domain and time domain (without proof)
Comparison to lumped single-segment model
Interconnect as an RC-tree
Delay and transition time representation
Transfer function of RC-tree and its properties. LP-filter
Moments and representation in s-domain.
Zero-pole and residua-pole representation. Dominant pole.
Elmore delay. Penfield-Rubinstein bounds. Elmore delay formula proof.
AWE. Delay metrics based on moments.
Transition time modeling
Elmore delay for general input
Central moments and their relation to moments
Representation of response as a probability distribution.
Alpert estimation for receiver slope. Delay and slope expressions based on moments
Explicit output slope calculation for singe RC-stage
Driver-receiver interaction
Driver modeling. K-equations
Admittance and impedance calculations
Effective capacitance and algorithm for its calculation.
Representation of load as “PI-model”.
Two-step delay approximation
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3. Recall – LTI system LTI: Linear Time Invariant
Linear: the response to linear combination of input signals produces linear combination of responses
Time Invariant: Shift of input signal in time causes the same shift of output signal
LTI system response is obtained by convolution:
where is impulse response

4. Recall – s-domain Laplace transform of the signal:
The Laplace transform of impulse response is system transfer function:
Fundamental relation between time-domain and s-domain:
Two-sided
One-sided (regular)

5. Basic input waveforms Dirac impulse (delta) function:
Heaviside step function:
Ramp function:
Saturated ramp function:

6. Laplace transforms of basic waveforms
Function
Laplace Transform

7. Interconnect modeling
output waveform
stage
input waveform
Need to predict waveform at the stage output(s), given waveform at the stage input
Usually it is enough to predict two main signal metrics instead of full waveform
signal delay
signal transition time

8. Delay and transition time
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Delay and transition time
Delay: 50% input to 50% output
Transition time (a.k.a slope, slew rate,
rise/fall time):
x% to 100-x% (on each signal)
Commonly used 20%-80% or 10%-90%
Mention that it is simple to define for monotonic waveforms, but much harder for non-monotonic
RC-line (without L) always produces monotonic waveform
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9. Delay and transition time calculation
Given input waveform
Given input transition time
Model input waveform with saturated ramp
Calculate output waveform :
Calculate transfer function
Calculate output in s-domain
Use inverse Laplace transform to calculate output in time domain
Solve
Delay =
Tr. time =

10. Delay and transition time calculation
Let us try it on two models for point-to-point interconnect:
Lumped RC-line
Distributed RC-line

11. Lumped model of RC-line
Ideal driver, no load
Transfer function:
For step input:

12. Distributed model of RC-line
Ideal driver, no load
How to derive transfer function?
Look at very small net segment and derive relations between voltage and current

13. Distributed model of RC-line
Heat transfer equation

14. Distributed model of RC-line
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Distributed model of RC-line
Boundary and initial conditions:
Solution (without proof):
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15. Distributed & lumped models of RC-line

16. Distributed model with driver and load
Voltage in Laplace domain:
Voltage in time domain (Sakurai’s approximation):
Time expressions:
“Closed-Form Expressions for Interconnection Delay, Coupling, and Crosstalk in VLSI's”, T. Sakurai, 1993

17. Interconnect tree However, real interconnect is an RC-tree
No capacitance between two nodes
No resistance between node and ground
It can be proven that such RC-tree is LTI system
In addition, such tree is a Low-Pass Filter

18. General interconnect General transfer function for interconnect:
Residue-pole representation:
Then impulse response is:
For unit step input:
The expression for saturated ramp is much more complex…
zeros
poles
residues

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Elmore delay
To calculate delay for unit step input we need to find so that
where is the response at output for steps input
Elmore proposal: “use centroid (mean) of impulse response instead”
median of impulse response
Hard to solve
mode
median
Somewhere should mention that accuracy of Elmore delay depends on spread (variance) – 2nd central moment and skew (assymetry) – 3rd central moment
mean
For RC-trees always:
“The transient response of damped linear networkswith particular regard to wide-band amplifiers”, W.C.Elmore, 1948
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20. Understanding Elmore delay
Moment representation of transfer function
Recall:
McLaurin expansion:
q-th circuit moment
- Elmore delay!

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הטכניון - מ.ט.ל. הפקולטה להנדסת חשמל - אביב תשס"ה
4:36:48 PM
Dominant pole metric
Since and , then
The Elmore delay:
Recall:
Therefore:
Show last line derivation explicitly
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Dominant pole metric
If there are no low-frequency zeros, then
If one of poles is dominant, i.e.
Then
Corresponding step response:
Think how to explain that “no low-frequency zeros”
Dominant pole 0.7TD can both overestimate and underestimate delay.
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Real Example
What is PRH?
Response at C5
Response at C1
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24. Calculating Elmore delay
Denote by output of the tree and by internal node
Define:
is the resistance of portion of the path between the input and
, what is common with the path between the input and node
is the resistance between input and output
is the resistance between input and node
Node k
Output i
Example:

25. Calculating Elmore delay
Voltage drop across the path from input to output:
current through capacitor
resistance contributing to voltage drop
Denote:
Node k
Output i

26. Calculating Elmore delay
On the other hand:
I.e., Elmore delay is the area above output voltage curve!
Therefore:
On the other hand,

27. Penfield & Rubinstein bounds
Function is used to derive lower and upper bounds for percent delay
“Signal Delay in RC tree Networks”, J. Rubinstein, P. Penfield, M. Horowitz, 1983

28. Asymptotic Waveform Evaluation (AWE)
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Asymptotic Waveform Evaluation (AWE)
Elmore delay uses first moment to represent system response
Not very accurate
For more accurate estimation, more moments are required
AWE uses moments matching to calculate parameters of low-order model:
where the reduced order is much less than the original order
AWE flow:
Generate moments from the circuit
Match the first moments to low-order pole model
Calculate residues
Obtain time-domain response by inverse Laplace transform
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29. Example: two-pole approximation
Assume we calcuated moments of the circuit:
Transfer function for reduced model:
McLauren expantion:
Coefficient match:

30. Example: two-pole approximation
The coefficients of denominator:
Residues are found similarly…
Time-domain response is given by

31. Backup

32. Some RC-tree system definitions
Input admittance:
Input impedance:
(voltage) Transfer function:

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הטכניון - מ.ט.ל. הפקולטה להנדסת חשמל - אביב תשס"ה
4:36:48 PM
Modeling distributed RC-line is too complicate even for single point-to-point line
Interconnect is usually represented by RC-tree
Easily computable and accurate
interconnect model is required
(delayout; slopeout)
(delayout; slopeout)
(delayin; slopein)
(delayout; slopeout)
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