1. Power Network Distribution Chung-Kuan Cheng CSE Dept. University of California, San Diego

2. Page  2 Power Distribution Network Overview  Background: power distribution networks (PDN’s)  Analysis: worst-case PDN noise prediction –Target Impedance –Worst Current Loads –Rogue Wave  Conclusions and future work

3. Page  3 Research on Power Distribution Networks  Analysis –Stimulus, Noise Margin, Simulation  Synthesis –VRM, Decap, ESR, Topology  Integration –Sensors, Prediction, Stability, Robustness

4. Page  4 Agenda  Background: power distribution networks (PDN’s)  Analysis: worst-case PDN noise prediction –Motivation –Problem formulation –Proposed Algorithm –Case study  Conclusions and future work

5. Page  5 Introduction: Motivation 5 Yeargte L nm freq GHz Vdd Volt PWPW I=P/V Amp Z=V/I Ohm 2011246.30.9390960.00964 2015178.50.811231520.00533 202010.712.40.681422080.00326 20247.416.60.601702840.00211 ITRS Roadmap: MPU Yeargte L nm freq GHz Vdd Volt PWPW I=P/V Amp Z=V/I Ohm 2011270.720.851.872.210.385 2015171.660.754.045.380.139 202010.73.310.657.7311.890.055 20247.45.320.6012.9221.530.028 SoC

6. Page  6 What is a power distribution network (PDN)  Power supply noise –Resistive IR drop –Inductive Ldi/dt noise [Popovich et al. 2008]

7. Page  7 Introduction  Target Impedance = V dd /I load x 5% –Production Cost  Negative Noise Budget –Negotiation between IC and package –Activity scheduling

8. Page  8 Agenda  Background: power distribution networks (PDN’s)  Analysis: worst-case PDN noise prediction –Motivation –Formulation –Algorithms –Case study  Conclusions and future work

9. Page  9 Analysis: Motivation  Target Impedance –Impedance in frequency domain  Worst power load in time domain –Slope of power load stimulus  Composite effect of resonance at multiple frequencies

10. Page  10 Target Impedance  PDN design –Objective: low power supply noise –Popular methodology: “target impedance” [Smith ’99] Implication: if the target impedance is small, then the noise will also be small

11. Page  11 Analysis: Formulation  Problems with “target impedance” design methodology –How to set the target impedance? Small target impedance may not lead to small noise –A PDN with smaller Z max may have larger noise  Time-domain design methodology: worst-case PDN noise –If the worst-case noise is smaller than the requirement, then the PDN design is safe. Straightforward and guaranteed –How to generate the worst-case PDN noise FT : Fourier transform

12. Page  12 Analysis: Related Work  At final design stages [Evmorfopoulos ’06] –Circuit design is fully or almost complete –Realistic current waveforms can be obtained by simulation –Problem: countless input patterns lead to countless current waveforms Sample the excitation space Statistically project the sample’s own worst-case excitations to their expected position in the excitation space  At early design stages [Najm ’03 ’05 ’07 ’08 ’09] –Real current information is not available –“Current constraint” concept –Vectorless approach: no simulation needed –Problem: assume ideal current with zero transition time

13. Page  13 Analysis: Formulation  Problem formulation I  PDN noise:  Worst-case current [Xiang ’09]: Zero current transition time. Unrealistic!

14. Page  14 Ideal Case Study: One-Stage LC Tank w/ ESR’s  Define:  Note  Under-damped condition:

15. Page  15 Ideal Case Study: One-Stage LC Tank w/ ESR’s (Cont’)  Step response: where  Normalized step response:

16. Page  16 Ideal Case Study: One-Stage LC Tank w/ ESR’s (Cont’)  Local extreme points of the step response:  Normalized magnitude of the first peak:

17. Page  17 Ideal Case Study: One-Stage LC Tank w/ ESR’s (Cont’)  Normalized worst-case noise:

18. Page  18 Ideal Case Study: One-Stage LC Tank w/ ESR’s (Cont’)  Impedance:  When [Mikhail 08]  Normalized peak impedance:

19. Page  19 Analysis: Algorithms  Problem formulation II T: chosen to be such that h(t) has died down to some negligible value. * f(t) replaces i(T-τ)

20. Page  20 Proposed Algorithm Based on Dynamic Programming  GetTransPos(j,k 1,k 2 ): find the smallest i such that F j (k 1,i)≤ F j (k 2,i)  Q.GetMin(): return the minimum element in the priority queue Q  Q.DeleteMin(): delete the minimum element in the priority queue Q  Q.Add(e): insert the element e in the priority queue Q

21. Page  21 Proposed Algorithm: Initial Setup  Divide the time range [0, T] into m intervals [t 0 =0, t 1 ], [t 1, t 2 ], …, [t m-1, t m =T]. h(t i ) = 0, i=1, 2, …, m-1  u 0 = 0, u 1, u 2, …, u n = b are a set of n+1 values within [0, b]. The value of f(t) is chosen from those values. A larger n gives more accurate results. h(t)

22. Page  22 Proposed Algorithm: f(t) within a time interval [ t j, t j+1 ]  I j (k,i) : worst-case f(t) starting with u k at time t j and ending with u i at time t j+1 h(t) Theorem 1: The worst-case f(t) can be cons- tructed by determining the values at the zero- crossing points of the h(t)

23. Page  23 Proposed Algorithm: Dynamic Programming Approach  Define V j (k,i) : the corresponding output within time interval [t j, t j+1 ]  Define the intermediate objective function OPT(j,i) : the maximum output generated by the f(t) ending at time t j with the value u i  Recursive formula for the dynamic programming algorithm:  Time complexity:

24. Page  24 Acceleration of the Dynamic Programming Algorithm  Without loss of generality, consider the time interval [t j, t j+1 ] where h(t) is negative.  Define W j (k,i) : the absolute value of V j (k,i) : Lemma 1 : W j (k 2,i 2 )- W j (k 1,i 2 )≤ W j (k 2,i 1 )- W j (k 1,i 1 ) for any 0 ≤ k 1 < k 2 ≤ n and 0 ≤ i 1 < i 2 ≤ n

25. Page  25 Acceleration of the Dynamic Programming Algorithm  Define F j (k,i) : the candidate corresponding to k for OPT(j,i)  Accelerated algorithm: –Based on Theorem 2 –Using binary search and priority queue Theorem 2: Suppose k 1 < k 2, i 1 ∈ [0,n] and F j (k 1,i 1 )≤ F j (k 2,i 1 ), then for any i 2 > i 1, we have F j (k 1,i 2 )≤ F j (k 2,i 2 ).

26. Page  26 Analysis: Case Study  Case 1: Impedance => Voltage drop –Transition Time  Case 2: Impedances vs Worst Cases  Case 3: Voltage drop due to resonance at multiple frequencies.

27. Page  27 Case Study 1: Impedance 2.09mΩ @ 19.8KHz 1.69mΩ @ 465KHz 3.23mΩ @ 166MHz

28. Page  28 Case Study 1: Impulse Response Impulse response: 100ns~10µs Impulse response: 10µs~100µs Impulse response: 0s~100ns High frequency oscillation at the beginning with large amplitude, but dies down very quickly Mid-frequency oscillation with relatively small amplitude. Low frequency oscillation with the smallest amplitude, but lasts the longest Amplitude = 1861 Amplitude = 29 Amplitude = 0.01

29. Page  29 Case Study 1: Worst-Case Current  Current constraints: Zoom in  The worst-case current also oscillates with the three resonant frequencies which matches the impulse response.  Saw-tooth-like current waveform at large transition times

30. Page  30 Case Study 1: Worst-Case Noise Response

31. Page  31 Case Study 1: Worst-Case Noise vs. Transition Time  The worst-case noise decreases with transition times.  Previous approaches which assume zero current transition times result in pessimistic worst-case noise.

32. Page  32 Case Study 2: Impedances vs Worst Cases 224.3KHz 11.2MHz 98.1MHz 224.3KHz 10.9MHz 101.6MHz

33. Page  33 Case Study 2: Worst-Case Noise  for both cases: meaning that the worst-case noise is larger than Z max.  The worst-case noise can be larger even though its peak impedance is smaller.

34. Page  34 Case 3: “Rogue Wave” Phenomenon  Worst-case noise response: The maximum noise is formed when a long and slow oscillation followed by a short and fast oscillation.  Rogue wave: In oceanography, a large wave is formed when a long and slow wave hits a sudden quick wave. Low-frequency oscillation corresponds to the resonance of the 2 nd stage High-frequency oscillation corresponds to the resonance of the 1 st stage

35. Page  35 Case 3: “Rogue Wave” Phenomenon (Cont’) Equivalent input impedance of the 2 nd stage at high frequency

36. Page  36 Case 3: “Rogue Wave” Phenomenon (Cont’)  Input current i(t): –Blue (I1): worst-case input stimulus –Red (I2): low frequency part of I1 –Green (I3): high frequency part of I1 I1=I2+I3

37. Page  37 Case 3: “Rogue Wave” Phenomenon (Cont’)  Input current i(t) (zoom in):

38. Page  38 Case 3: “Rogue Wave” Phenomenon (Cont’)  Noise response @ chip output –Blue (V1): response of I1 –Red (V2): response of I2 –Green (V3): response of I3

39. Page  39 Case 3: “Rogue Wave” Phenomenon (Cont’)  Noise response (zoom in):

40. Page  40 Remarks  Worst-case PDN noise prediction with non-zero current transition time –The worst-case PDN noise decreases with transition time –Small peak impedance may not lead to small worst-case noise –“Rogue wave” phenomenon  Adaptive parallel flow for PDN simulation using DFT –0.093% relative error compared to SPICE –10x speed up with single processor. –Parallel processing reduces the simulation time even more significantly

41. Page  41 Summary 1.Throughput/power (instruction/energy) 2.Throughput 2 /power (f x instruction/energy)  Power Distribution Network –VRMs, Switches, Decaps, ESRs, Topology,  Analysis –Stimulus, Noise Tolerance, Simulation  Control (smart grid) –High efficiency, Real time analysis, Stability, Reliability, Rapid recovery, and Self healing

42. Page  42

43. Page  43 Publication List Power Distribution Network Simulation and Analysis [1] W. Zhang and C.K. Cheng, "Incremental Power Impedance Optimization Using Vector Fitting Modeling,“ IEEE Int. Symp. on Circuits and Systems, pp. 2439-2442, 2007. [2] W. Zhang, W. Yu, L. Zhang, R. Shi, H. Peng, Z. Zhu, L. Chua-Eoan, R. Murgai, T. Shibuya, N. Ito, and C.K. Cheng, "Efficient Power Network Analysis Considering Multi-Domain Clock Gating,“ IEEE Trans on CAD, pp. 1348-1358, Sept. 2009. [3] W.P. Zhang, L. Zhang, R. Shi, H. Peng, Z. Zhu, L. Chua-Eoan, R. Murgai, T. Shibuya, N. Ito, and C.K. Cheng, "Fast Power Network Analysis with Multiple Clock Domains,“ IEEE Int. Conf. on Computer Design, pp. 456-463, 2007. [4] W.P. Zhang, Y. Zhu, W. Yu, R. Shi, H. Peng, L. Chua-Eoan, R. Murgai, T. Shibuya, N. Ito, and C.K. Cheng, "Finding the Worst Case of Voltage Violation in Multi-Domain Clock Gated Power Network with an Optimization Method“ IEEE DATE, pp. 540-547, 2008. [5] X. Hu, W. Zhao, P. Du, A.Shayan, C.K.Cheng, “An Adaptive Parallel Flow for Power Distribution Network Simulation Using Discrete Fourier Transform,” IEEE/ACM Asia and South Pacific Design Automation Conference (ASP-DAC), 2010. [6] C.K. Cheng, P. Du, A.B. Kahng, G.K.H. Pang, Y. Wang, and N. Wong, "More Realistic Power Grid Verification Based on Hierarchical Current and Power Constraints,“ ACM Int. Symp. on Physical Design, pp. 159-166, 2011.

44. Page  44 Publication List Power Distribution Network Analysis and Synthesis [7] W. Zhang, Y. Zhu, W. Yu, A. Shayan, R. Wang, Z. Zhu, C.K. Cheng, "Noise Minimization During Power-Up Stage for a Multi-Domain Power Network,“ IEEE Asia and South Pacific Design Automation Conf., pp. 391-396, 2009. [8] W. Zhang, L. Zhang, A. Shayan, W. Yu, X. Hu, Z. Zhu, E. Engin, and C.K. Cheng, "On-Chip Power Network Optimization with Decoupling Capacitors and Controlled-ESRs,“Asia and South Pacific Design Automation Conference, 2010. [9] X. Hu, W. Zhao, Y.Zhang, A.Shayan, C. Pan, A. E.Engin, and C.K. Cheng, “On the Bound of Time-Domain Power Supply Noise Based on Frequency-Domain Target Impedance,” System Level Interconnect Prediction Workshop (SLIP), July 2009. [10] A. Shayan, X. Hu, H. Peng, W. Zhang, and C.K. Cheng, “Parallel Flow to Analyze the Impact of the Voltage Regulator Model in Nanoscale Power Distribution Network,” In. Symp. on Quality Electronic Design (ISQED), Mar. 2009. [11] X. Hu, P. Du, and C.K. Cheng, "Exploring the Rogue Wave Phenomenon in 3D Power Distribution Networks,“ IEEE Electrical Performance of Electronic Packaging and Systems, pp. 57-60, 2010. [12] C.K. Cheng, A.B. Kahng, K. Samadi, and A. Shayan, "Worst-Case Performance Prediction Under Supply Voltage and Temperature Variation,“ ACM/IEEE Int. Workshop on System Level Interconnect Prediction, pp. 91-96, 2010.

45. Page  45 Publication List (Cont’) 3D Power Distribution Networks [13] A. Shayan, X. Hu, “Power Distribution Design for 3D Integration”, Jacob School of Engineering Research Expo, 2009 [Best Poster Award] [14] A. Shayan, X. Hu, M.l Popovich, A.E. Engin, C.K. Cheng, “Reliable 3D Stacked Power Distribution Considering Substrate Coupling”, in International Conference on Computre Design (ICCD), 2009. [15] A. Shayan, X. Hu, C.K. Cheng, “Reliability Aware Through Silicon Via Planning for Nanoscale 3D Stacked ICs,” in Design, Automation & Test in Europe Conference (DATE), 2009. [16] A. Shayan, X.g Hu, H. Peng, W. Zhang, C.K. Cheng, M. Popovich, and X. Chen, “3D Power Distribution Network Co-design for Nanoscale Stacked Silicon IC,” in 17 th Conference on Electrical Performance of Electronic Packaging (EPEP), Oct. 2008. [5] [17] W. Zhang, W. Yu, X. Hu, A.i Shayan, E. Engin, C.K. Cheng, "Predicting the Worst-Case Voltage Violation in a 3D Power Network", Proceeding of IEEE/ACM International Workshop on System Level Interconnect Prediction (SLIP), 2009. [18] X. Hu, P. Du, and C.K. Cheng, "Exploring the Rogue Wave Phenomenon in 3D Power Distribution Networks,“ IEEE Electrical Performance of Electronic Packaging and Systems, pp. 57-60, 2010.