PV-PPV: Parameter Variability Aware, Automatically Extracted, Nonlinear Time-Shifted Oscillator Macromodels
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1 PV-PPV: Parameter Variability Aware, Automatically Extracted, Nonlinear Time-Shifted Oscillator Macromodels Zhichun Wang, Xiaolue Lai and Jaijeet Roychowdhury Dept of ECE, University of Minnesota, Twin Cities 1
2 Outline Parameter Variations and their Impact Difficulties of Variability Simulation Oscillators Simulation of Oscillators Variability Simulation of Oscillators Our contribution: Parameterized PPV Macromodels (PV- PPV) Validation 2
3 Parameter Variations and their Impact 3
4 Impact of Process Variations Shekhar Borkar, Circuit Research Lab, Intel 180nm CMOS technology 20x variation in chip leakage 30% variation in chip freq Affects yield Distribution of frequency and standby leakage current of microprocessors in a wafer 4
5 Typical Parameters of Interest Supply voltage variations Depend on, e.g., supply network parasitics inductance, capacitance, resistance Impact: e.g., 10% VDD variation causes 20% delay Temperature variations Dynamic variation Direct impact on max reliable frequency Parameter Variability Analysis is a MUST in Circuit and Microarchitecture Design 5
6 Need for Variability Simulation Variability Simulation This work fits here Circuit with N varying parameters Yield, statistics or bounds of performance Design Centering 6
7 Parameter Variability Simulation can be EXPENSIVE Variability Simulation Very Important Circuit with N varying parameters Yield, statistics or bounds of performance Design Centering 7
8 Parameter Variability Simulation can be EXPENSIVE Involves Variability thousands Simulation of simulations! Very Important Worst-case corner analysis samples combinations of parameters The number of combinations is huge e.g., 2 N for min/max bounds 8
9 Oscillators 9
10 Oscillators in Electronic Systems Signal carriers in communication systems VCOs, frequency dividers, PLLs for mixed-signal, high-speed digital, etc. Phase-locked loop 10
11 Oscillator: A Special Simulation Challenge SPICE-based transient simulation Computation/size/accuracy: much greater than for amps/mixers fundamental property of all oscillators numerical errors in phase keep increasing tiny timesteps needed per cycle inefficient for even 1-transistor oscillators long startups: many cycles integrated RF: 100s to 1000s of transistors SPICE-based transient simulation of oscillators is not a good idea 11
12 Oscillator Macromodels Abstraction (manual/ automated) Small system or circuit fewer equations much easier to solve Macromodels need to be appropriate for: nature of circuit (e.g., oscillator) which performance is of interest (e.g., phase, amplitude) In oscillators, phase is of key interest 12
13 Phase Shifts in Oscillators Cross-coupled CMOS osc Node Voltage Oscillation waveform of node voltage Waveform under external input External input time Phase shift function of time Phase macromodels directly solve for phase shifts 13
14 Oscillator PPV Phase Macromodels Efficient automated extraction Oscillator PPV phase macromodels node voltages, branch currents large system, many equations scalar equation 14
15 The PPV Phase Macromodels Single scalar differential equation Phase shift nonlinear Perturbation Projection Vector (PPV) Speedup: 100x to 1000x or more over transient simulation depends on circuits and applications External inputs automated extracted 15
16 How PPV Macromodels are Used Replace oscillator with PPV macromodels 16
17 How PPV Macromodels are Used Replace oscillator with PPV macromodels phase calculated by the PPV macromodel waveforms of node voltages or branch currents Node Voltage Oscillation waveforms Phase shifts Small amplitude variations PPV Macromodels time waveforms of node voltages or branch currents 17
18 Variability Simulation of Oscillators 18
19 Variability Simulation using PPV Macromodels Circuit with N varying parameters PPV macromodel Involves Automated full circuit HB simulation, computationally PPV extraction expensive! Repeated when parameters change Only once Parameterized PPV extraction PPV macromodel based simulation Parameterized PPV macromodel (PV-PPV) 19
20 PV-PPV: Parameterized PPV Macromodels ODE with both Perturbation and Parameter Variation Parametrized PPV Macromodels Also solve for the phase shift 20
21 PV-PPV: Parameterized PPV Macromodels (cont.) New term for variability: like a time-varying input Captures nonlinearity with respect to parameter variations Changed for different Extra computation is trivial parameter set 21
22 PV-PPV: Advantages and Limitations Advantages: Avoids re-extraction of PPV macromodels when parameters change especially useful for circuits with many coupled oscillators (e.g., high-speed serialized I/O: HyperTransport, PCI Express) Don't need original circuit for parameter variability simulation for intellectual property (IP) protection Limitations: Linearization used: only applies to small parameter variations but good enough for most applications 22
23 Validation 23
24 Validation Metrics Oscillator frequency changes due to parameter changes (no external input) validate accuracy, speedups (compared to repeated extraction via HB) System application: many coupled oscillators network of 40,000 coupled oscillators with randomly varying parameters 24
25 Speedups for Calculating Center Frequency Shifts (compared to HB) Using HB simulation many HB simulations: one per parameter set (time consuming) Using PV-PPV only one HB simulation: at nominal parameter values (expense of repeated HB avoided) one extraction: PV-PPV macromodel (efficient) one PPV-based transient simulation per parameter set (scalar equation, very fast) Speedups depend on number of parameter sets of interest for simulation 25
26 3-Stage Ring Oscillator Nominal parameters varying parameter Nominal frequency 26
27 Center Frequency vs Threshold Voltage Relative frequency change (PV-PPV vs HB) Relative frequency error (compared to HB) 33.8sec Speedup: 7.04x for 20 samples 238sec Nonlinearity wrt inductance captured by PV-PPV Error: 0.07% 27
28 Cross-Coupled LC Oscillator Nominal parameters Nominal frequency varying parameter 28
29 Center Frequency vs Inductance Relative frequency change (PV-PPV vs HB) Relative frequency error (compared to HB) 86.5sec Speedup: 11.7x for 20 samples 16.9min Still good for some application 29
30 Injection Locking in Oscillators 30
31 System Simulation: Many Coupled Oscillators with Varying Parameters Oscillator Coupling Many oscillators Different parameters for each oscillator Where do such systems arise? 31
32 Reasons for System Simulation Many ring oscillators on a single wafer new process Courtesy Sani Nassif, IBM; thanks also to TM Mak, Intel variability measurements undesired coupling Bio-mimetic collaborative radio deliberately designed coupling 32
33 3x3 sub array 3-stage ring oscillator Each blue line: a resistor 33
34 Pattern Formation in 200x200 Coupled Oscillator Network coupled oscillators Each little square: an oscillator Color: phase of each oscillator Many oscillators with same phase Spontaneous pattern formation 34
35 Pattern Formation in 200x200 Coupled Oscillator Network No parameter variations Gaussianly distributed Vth Imperfection due to parameter variations 35
36 Speedups for Simulating Networks of Many Coupled Oscillators coupled oscillators each oscillator: 3-stage ring (BSIM3 MOS model) For 200x200 network of locally coupled oscillators: CPU time (AMD Athlon 64 Dual Core 3800+; 1GB RAM): extracting one PV-PPV macromodel: 72.7s extracting 200x200 PPV macromodels: 3.36 days Speedup: 39932x 36
37 Summary Parameterized PPV macromodels (PV-PPV) avoid re-extraction of PPV macromodels for different parameters intellectual property (IP) protection capture nonlinear effects Speedups: calculating frequency shifts for 20 parameters 7.04x (ring oscillator), 11.7x (LC oscillator) 39932x for simulating coupled oscillators 37
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