1. Chapter 4 Interconnect Analysis

2. 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

3. 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

4. Chapter 4.1 Linear System Laplace Transformation Pole/residue Basic Circuit Analysis

5. Laplace Transformation Definition : time domainfrequency domain Time domain (t domain) Complex frequency domain (s domain) Linear Circuit Differential equation Response waveform Laplace Transform Inverse Transform Linear equation Response transform L L -1

6. Frequency Domain Transfer Function and Time Domain Impulse Response Frequency domain representation H(s) u(s)y(s) = H(s) u(s) Linear system h(t) u(t) Linear system Time domain representation The transfer function H(s) is the Laplace Transform of the impulse response h(t)

7. Circuit Analysis Using Laplace Transforms Time domain (t domain) Complex frequency domain (s domain) Linear Circuit Differential equation Classical techniques Response waveform Laplace Transform Inverse Transform Algebraic equation Algebraic techniques Response transform L L -1

8. Poles and Zeros of F(s) Scale factor: K = b m /a n Poles: s = p k (k = 1, 2,..., n) Zeros: s = z k (k = 1, 2,..., m) Resonant frequencies

9. Pole-Zero Diagrams s-plane pole location zero location s-plane

10. Poles and Waveforms If poles in right-plane, waveform increases without bound 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

11. 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 v N (t) (zero-input response) Forced response v F (t) (zero-state response) For linear circuits:

12. unit step function u(t)= 0 1 1 pulse function of width T 0 1/ T -T/2T/2 unit impulse function Inputs

13. Time Moments of Impulse Response h(t) Definition of moments i-th moment

14. Chapter 4.2 Elmore Delay Lumped and distributed interconnect delay model Elmore delay and distributed interconnect delay model Elmore delay and time moments

15. Interconnect Model Lumped vs Distributed LumpedDistributed R C r c r c r c r c

16. Analysis of Simple RC Circuit zero-input response: (natural response) step-input response: match initial state: output response for step-input: v0v0 v 0 u(t) v 0 (1-e RC/T )u(t)

17. RC-Tree –The network has a single input node –All capacitors between node and ground –The network does not contain any resistive loop R1R1 C1C1 s R2R2 C2C2 R4R4 C4C4 C3C3 R3R3 CiCi RiRi 1 2 3 4 i

18. RC-tree Property –Unique resistive path between the source node s and any other node i of the network  path resistance R ii Example: R 44 =R 1 +R 3 +R 4 R1R1 C1C1 s R2R2 C2C2 R4R4 C4C4 C3C3 R3R3 CiCi RiRi 1 2 3 4 i

19. RC-tree Property –Extended to shared path resistance R ik : Example:R i4 =R 1 +R 3 R i2 =R 1 R1R1 C1C1 s R2R2 C2C2 R4R4 C4C4 C3C3 R3R3 CiCi RiRi 1 2 3 4 i

20. 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

21. RC-chain (or ladder) Special case Shared-path resistance  path resistance  R1R1 C1C1 R2R2 C2C2 RNRN CNCN V in VNVN

22. RC-Line Delay R C R C R C V in VNVN 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

23. Time Moments of Impulse Response h(t) Definition of moments i-th moment Note that m 1 = Elmore delay when h(t) is monotone voltage response of impulse input

24. Elmore Delay for RC Trees Definition –h(t) = impulse response –T D = mean of h(t) = Interpretation –H(t) = output response (step process) –h(t) = rate of change of H(t) –T 50% = median of h(t) –Elmore delay approximates the median of h(t) by the mean of h(t) median of v’(t) (T 50% ) h(t) = impulse response H(t) = step response

25. Elmore Delay in RC Tree input i k jSiSi path resistance R ii R jk

26. Proof of Theorem