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

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

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 Prepared by Lei He

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 Prepared by Lei He

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 Prepared by Lei He

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 Prepared by Lei He

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 Prepared by Lei He

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 Prepared by Lei He

Time Moments of Impulse Response h(t) ECE902 VLSI Interconnect Time Moments of Impulse Response h(t) Definition of moments i-th moment Prepared by Lei He

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

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? Prepared by Lei He

Lumped RC Model R v(t) u(t) C Impulse response and step response of a lumped RC circuit

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

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

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

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 Prepared by Lei He

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 Prepared by Lei He

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 Prepared by Lei He

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 Prepared by Lei He

ECE902 VLSI Interconnect Proof of Theorem Prepared by Lei He

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) Prepared by Lei He

Elmore Delay Approximation Elmore delay approximates 50% delay

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 Prepared by Lei He

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

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

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

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

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:

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.

S2P Vs. Elmore Delay

Reading Assignment [1] Elmore delay model http://eda.ee.ucla.edu/EE201A-04Spring/elmore.pdf [2] Elmore delay formula for RC tree http://eda.ee.ucla.edu/EE201A-04Spring/Elmore_TCAD.pdf [3] S2P algorithm Upload them to wiki

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