AMultilevel Technique forRobust andEfficient Extraction ofPhaseMacromodels ofDigitally Controlled Oscillators

integration issues, digital phase-locked loop (DPLL) design techPPVphase macromodels areimportant forspeeding upsimulationniques [6-8] haverapidly gained inpopularity, especially forhighlyofoscillator related circuits, suchasPLLs, without sacrificing ac- integrated SoCapplications. DPLLsprovide characteristics such curacy. Prior numerical methods forextracting PPVsface verysig- asfast switching, full digitization, andgoodfrequency stability [9]. nificant robustness andaccuracy problems whenconfronted with Thecorecomponent ofaDPLL istheswitch-tuned digitally condigitally controlled oscillators (DCOs, corebuilding blocks indigi- trolled oscillator (DCO)[10, 11], thedigital counterpart ofanalog talphase-locked loops), duetolarge RCtime-constants fromgated VCOs. Atypical DCOconsists ofmanygated capacitors orresiscapacitors. Inthis paper, wepresent ahierarchical harmonic bal- in Figure tr ancebased technique fornumerically extracting thePPVofDCOs inFigure 1. fromtheir SPICE-level circuit descriptions. Theproposed method However, theadvent ofDCOsandDPLLsposes unique chalapplies hierarchical circuit partitioning andmulti-level Newton meth- lenges forsimLlation andverification, already adifficult enough odstoachieve dramatically superior convergence andPPVaccu- task forPLLs.SPICE-level getransient" simulation [12] ispoorly racyinthepresence oflarge RCtime-constants. Wevalidate the suited forsimulating PLLsingeneral, duetoefficiency issues [17]. method onalarge DCOwithmanygated capacitors anddemon- Toalleviate this inefficiency, apopular approach istoperform apstrate that itcanextract thePPVefficiently androbustly, succeed- proximate PLLsimulation using phasedomain macromodelling of ingwhenprior methods fail. Themethod also provides speedups sections ofthePLL's feedback loop, including theVCO.Various ofanorder ofmagnitude forlarge circuits, inaddition tohaving phase domain macromodels foroscillators (e.g. [13-16]) areavailsignificantly smaller memoryrequirements. able; amongthese, theperturbation projection vector (PPV)[16] model, anonlinear time-shifted macromodel that canbeextracted fromSPICEnetlists using numerical methods, hasbeenshownto