Coupled oscillator networks occur in various domains such as biology, astrophysics and electronics. In this paper, we present a comprehensive procedure for rapid and accurate simulation of large coupled oscillator networks using widely accepted, fully-nonlinear perturbation projection vector (PPV) phase macromodels. We validate our method against full simulation of 20times20 coupled network of Brusselator biochemical oscillator and obtain computational speedups of 170x over full simulation. Furthermore, we apply the method to study self-organization phenomenon of Brusselator under asymmetric coupling and time period variations.

Perturbation projection vector (PPV) is an established technique for oscillator phase noise analysis; However, the PPV method significantly loses accuracy when circuits have large time constants, resulting in over-estimation of oscillator phase noise. In this paper, we show the problem of the PPV phase equation that it ignores the dynamics of oscillator frequency responses, and propose an improved PPV technique: Frequency-Aware PPV (FW-PPV). Using the FW-PPV technique, we derive analytical phase noise equations, which work accurately for all oscillators, especially for oscillators with large time constants. We apply the proposed technique on real oscillator circuits, and compare the results to Cadence Spectre RF. Simulation results show that our method has better accuracy than previous approaches.

We present a robust, automated oscillator macromodeling technique for extracting comprehensive phase and amplitude macromodels from oscillators' SPICE circuit descriptions. The macromodels are able to correctly predict oscillator response in the presence of interference at far lower computational cost than that of full SPICE-level simulation, while retaining the simulation accuracy. We find many applications for the proposed macromodeling technique, which include injection locking prediction, fast simulation of coupled oscillating systems, and fast PLL transient simulation and jitter analysis. We demonstrate the applications on some oscillator-based systems, and compare results against full SPICE-level simulation. Experimental results show that the macromodels capture the behavior of the oscillator systems accurately, and provide speedups of over three orders of magnitude.

The PPV is a robust phase domain macromodel for oscillators. It has been proven to predict oscillators' responses correctly under small signal perturbations, and capture nonlinear phase effects such as injection locking/pulling. In this work, we present a novel approach to extend the PPV macromodel to handle variability in circuit parameters. We derive a modified PPV-based phase equation in which parameter variations are modelled as special inputs. An important feature of our technique is that it avoids PPV re-extraction, this resulting in great convenience and efficiency in its use for, e.g., Monte Carlo type simulations. Using LC and ring oscillators as examples, we demonstrate the capability of the proposed technique for capturing parameter variation effects in injection locking analysis. Simulation results show that our new approach accurately predicts the maximum locking range of oscillators with speedups of two orders of magnitude over direct simulation.

We present a novel methodology suitable for fast, correct design of modern PLLs. The central feature of the methodology is its use of accurate, nonlinear behavioral models for the VCO within the PLL, thus removing the need for many time-consuming SPICE-level simulations during the design process. We apply the new methodology to design a novel injection-aided PLL that acquires lock 3/spl times/ faster than prior designs, without trading off other design metrics such as jitter. We demonstrate how existing design methodologies based on behavioral simulation are incapable of leading to our new PLL design. The nonlinear behavioral simulations employed in our methodology are about 2 orders of magnitude faster than transistor-level ones, resulting in an overall design productivity gain of an order of magnitude.

We address the problem of fast and accurate computational analysis of large networks of coupled oscillators arising in nanotechnological and biochemical systems. Such systems are computationally and analytically challenging because of their very large sizes and the complex nonlinear dynamics they exhibit. We develop and apply a nonlinear oscillator macromodel that generalizes the well-known Kuramoto model for interacting oscillators, and demonstrate that using our macromodel provides important qualitative and quantitive advantages, especially for predicting self-organization phenomena such as spontaneous pattern formation. Our approach extends and applies recently-developed computational methods for macromodel ling electrical oscillators, and features both phase and amplitude components that are extracted automatically (using numerical algorithms) from more complex differential-equation oscillator models available in the literature. We apply our approach to networks of tunneling phase logic (TPL) and Brusselator biochemical oscillators, predicting a variety of spontaneous pattern generation phenomena. Comparing our results with published measurements of spiral, circular and other pattern formation, we show that we can predict these phenomena correctly, and also demonstrate that prior models (like Kuramoto's) cannot do so. Our approach is more than 3 orders of magnitude faster than techniques that are comparable in accuracy.

We present an efficient method for automatically extracting unified amplitude/phase macromodels of arbitrary oscillators from their SPICE-level circuit descriptions. Such comprehensive oscillator macromodels are necessary for accuracy when speeding up simulation of higher-level circuits/systems, such as PLLs, in which oscillators are embedded. Standard MOR techniques for linear time invariant (LTI) and varying (LTV) systems are not applicable to oscillators on account of their fundamentally nonlinear phase behavior. By employing a cancellation technique to deflate out the phase component, we restore the validity and efficacy of Krylov-subspace-based LTV MOR techniques for macromodelling oscillator amplitude responses. The nonlinear phase response is re-incorporated into the macromodel after the amplitude components have been reduced. The resulting unified macromodels predict oscillator waveforms, in the presence of any kind of input or interference, at far lower computational cost than full SPICE-level simulation, and with far greater accuracy compared to existing macromodels. We demonstrate the proposed techniques on LC and ring oscillators, obtaining speedups of 30-120/spl times/ with no appreciable loss of accuracy, even for small circuits.

PPV phase macromodels are important for speeding up simulation of oscillator related circuits, such as PLLs, without sacrificing accuracy. Prior numerical methods for extracting PPVs face very significant robustness and accuracy problems when confronted with digitally controlled oscillators (DCOs, core building blocks in digital phase-locked loops), due to large RC time-constants from gated capacitors. In this paper, we present a hierarchical harmonic balance based technique for numerically extracting the PPV of DCOs from their SPICE-level circuit descriptions. The proposed method applies hierarchical circuit partitioning and multi-level Newton methods to achieve dramatically superior convergence and PPV accuracy in the presence of large RC time-constants. We validate the method on a large DCO with many gated capacitors and demonstrate that it can extract the PPV efficiently and robustly, succeeding when prior methods fail. The method also provides speedups of an order of magnitude for large circuits, in addition to having significantly smaller memory requirements.

We present a novel method for generating small, accurate PLL macromodels that capture transient response and jitter performance with unprecedented accuracy, while offering large speedups. The method extracts and uses a highly accurate oscillator phase macromodel termed the TP-PPV macromodel. The core idea behind the novel extraction procedure is to combine concepts from strongly nonlinear trajectory piecewise macromodeling techniques together with PPV-based timeshifted nonlinear phase macromodels. As a result, TP-PPV generated macromodels offer excellent global as well as local fidelity. These properties are necessary for handing large excursions in PLL control voltages during capture/lock in, e.g., hopping frequency synthesizers. We validate TP-PPV on a 5-stage interpolative ring VCO based PLL and compare results against full simulation, as well as against prior macromodels. We show that, unlike prior macromodels that only work well when the control voltage of the VCO has small excursions, the TP-PPV macromodel provides near-perfect matches against full SPICElevel simulation over a wide range of design scenarios, while achieving speedups of about three orders of magnitude.

Phase-locked loops (PLLs) are widely used in electronic systems. As PLL malfunction is one of the most important factors in re-fabs of SoCs, fast simulation of PLLs to capture non-ideal behavior accurately is an immediate, pressing need in the semiconductor design industry. In this paper, we present a nonlinear macromodel based PLL simulation technique that is considerably more accurate than prior linear PLL simulation techniques. Our method is able to accurately capture transient behavior and faithfully estimate timing jitter in noisy PLLs. We demonstrate the proposed technique on ring and LC voltage-controlled oscillator (VCO) based PLLs, and compare results against linear PLL macromodels and full SPICE-level simulation. We show that, unlike prior linear macromodel based approaches, the proposed nonlinear technique captures the dynamics of complex phenomena such as locking, cycle slipping and power supply noise induced PLL jitter, replicating qualitative features from full SPICE simulations accurately while providing speedups of over two orders of magnitude.

We present a method for extracting comprehensive amplitude and phase macromodels of oscillators from their circuit descriptions. The macromodels are based on combining a scalar, nonlinear phase equation with a small linear time-varying system to capture slowly-dying amplitude variations. The comprehensive macromodels are able to correctly predict oscillator response in the presence of interference at far lower computational cost than that of full SPICE-level simulation. We also present an efficient numerical method for capturing injection locking in oscillators, thereby improving on the classic technique of Adler (1946) in terms of accuracy and applicability to any kind of oscillator. We demonstrate the proposed techniques on LC and ring oscillators, comparing results from the macromodels against full SPICE-like simulation. Numerical experiments demonstrate speed tips of orders of magnitude, while retaining excellent accuracy.