Interconnect Analysis for Linear Systems in Electrical Engineering

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

2. 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

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

4. Resonant frequencies 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)

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

6. If poles in right-plane, waveform increases without bound as time approaches infinity Complex poles come in pairs that produce oscillatory waveforms Real poles produce exponential waveforms If poles in left-plane, waveform decays to zero as time approaches infinity If poles on j-axis, waveform neither decays nor grows Poles and Waveforms

7. Basic Circuit Analysis • Basic waveforms • Step input • Pulse input • Impulse Input • Use simple input waveforms to understand the impact of network design

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

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

10. Organization • Linear System • First Order Analysis (Elmore Delay) • Second Order Analysis

11. R r r r r c c c c C Interconnect ModelLumped vs Distributed Lumped Distributed How to analyze the delay for each model?

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

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

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

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

16. 2 R4 R2 C4 C2 R1 4 1 s R3 Ri Ci C1 3 C3 i Distributed RC-Tree • The network has a single input node • All capacitors between node and ground • The network does not contain any resistive loop

17. 2 R4 R2 C4 C2 R1 4 1 s R3 Ri Ci C1 3 C3 i RC-tree Property • Unique resistive path between the source node s and any other node i of the network  path resistance Rii Example: R44=R1+R3+R4

18. 2 R4 R2 C4 C2 R1 4 1 s R3 Ri Ci C1 3 C3 i RC-tree Property • Extended to shared path resistance Rik: Example: Ri4=R1+R3 Ri2=R1

19. 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

20. Proof of Theorem

21. 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) h(t) = impulse response H(t) = step response median of v’(t) (T50%) Interpretation of Elmore Delay

22. Elmore Delay Approximation Elmore delay approximates 50% delay

23. R1 R2 RN C1 C2 CN Vin VN RC-chain (or ladder) • Special case • Shared-path resistance path resistance

24. R R R C C C RC-Chain Delay VN Vin 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

25. Organization • Linear System • First Order Analysis (Elmore Delay) • Second Order Analysis

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* andm1* are the moments of H(s). m0*isthe Elmore delay.

30. S2P Vs. Elmore Delay

31. [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 Reading Assignment

32. [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 hw2