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 049031 Lecture 3: Interconnect modeling Avinoam Kolodny Konstantin Moiseev VLSI-מודלים ואופטימיזציה של קווי חיבור ב

הטכניון - מ.ט.ל. הפקולטה להנדסת חשמל - אביב תשס"ה הטכניון - מ.ט.ל. הפקולטה להנדסת חשמל - אביב תשס"ה 4:36:48 PM 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 VLSI-מודלים ואופטימיזציה של קווי חיבור ב

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

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)

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

Laplace transforms of basic waveforms Function Laplace Transform

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

Delay and transition time הטכניון - מ.ט.ל. הפקולטה להנדסת חשמל - אביב תשס"ה 4:36:48 PM 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 VLSI-מודלים ואופטימיזציה של קווי חיבור ב

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 =

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

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

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

Distributed model of RC-line Heat transfer equation

Distributed model of RC-line הטכניון - מ.ט.ל. הפקולטה להנדסת חשמל - אביב תשס"ה 4:36:48 PM Distributed model of RC-line Boundary and initial conditions: Solution (without proof): VLSI-מודלים ואופטימיזציה של קווי חיבור ב

Distributed & lumped models of RC-line

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

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

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

הטכניון - מ.ט.ל. הפקולטה להנדסת חשמל - אביב תשס"ה הטכניון - מ.ט.ל. הפקולטה להנדסת חשמל - אביב תשס"ה 4:36:48 PM 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 VLSI-מודלים ואופטימיזציה של קווי חיבור ב

Understanding Elmore delay Moment representation of transfer function Recall: McLaurin expansion: q-th circuit moment - Elmore delay!

הטכניון - מ.ט.ל. הפקולטה להנדסת חשמל - אביב תשס"ה 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 VLSI-מודלים ואופטימיזציה של קווי חיבור ב

הטכניון - מ.ט.ל. הפקולטה להנדסת חשמל - אביב תשס"ה הטכניון - מ.ט.ל. הפקולטה להנדסת חשמל - אביב תשס"ה 4:36:48 PM 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. VLSI-מודלים ואופטימיזציה של קווי חיבור ב

הטכניון - מ.ט.ל. הפקולטה להנדסת חשמל - אביב תשס"ה הטכניון - מ.ט.ל. הפקולטה להנדסת חשמל - אביב תשס"ה 4:36:48 PM Real Example What is PRH? Response at C5 Response at C1 VLSI-מודלים ואופטימיזציה של קווי חיבור ב

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:

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

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

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

Asymptotic Waveform Evaluation (AWE) הטכניון - מ.ט.ל. הפקולטה להנדסת חשמל - אביב תשס"ה 4:36:48 PM 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 VLSI-מודלים ואופטימיזציה של קווי חיבור ב

Example: two-pole approximation Assume we calcuated moments of the circuit: Transfer function for reduced model: McLauren expantion: Coefficient match:

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

Backup

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

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 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) VLSI-מודלים ואופטימיזציה של קווי חיבור ב