Alaeddin Aydiner

A silicon-validated methodology for power delivery modeling and simulation

Power integrity has become increasingly important for the designs in 32nm or below. This paper discusses a silicon-validated methodology for microprocessor power delivery modeling and simulation. There have been many prior works focusing on power delivery analysis and optimization. However, none of them provided a comprehensive modeling methodology with post-silicon data to validate the use of the models. In this paper, we present power delivery system models that are able to achieve less than 10% deviation from the supply noise measurements on a 32nm industrial microprocessor design. Our models are able to capture the unique impacts of on-die inductance, state dependent coupling capacitance and die-package interaction. Those impacts happen to be prominent for the designs in 32nm or below but were considered negligible or even not noted in earlier technology nodes. Comparisons were made to quantify the impacts of different modeling strategies on supply noise prediction accuracy. This specifically provides designers insights in selecting appropriate models for power delivery analysis. The impact of power delivery noise on timing margin was accurately estimated showing a good agreement to the worst-case jitter measurements.

An Enhanced Augmented Electric-Field Integral Equation Formulation for Dielectric Objects

A full-wave surface integral equation (SIE) method based on the augmented electric-field integral equation (A-EFIE) for dielectric objects with low-frequency stability is presented in this paper. Motivated by the A-EFIE formulation for perfect electric conductor (PEC), the internal and external problems are both augmented with the current continuity equation and renormalized to eliminate the low-frequency breakdown. Although the magnetic-field integral equation operator K is free of low-frequency breakdown, its matrix form is ill-conditioned and unsolvable if the traditional Rao-Wilton-Glisson (RWG) basis function is used as the testing and basis functions. As a remedy, the Buffa-Christiansen (BC) basis function is introduced to alleviate this testing issue. After this treatment, the matrix form of operator K is well conditioned. To solve problems with a large number of unknowns, a preconditioning scheme is introduced to accelerate the convergence and the mixed-form fast multipole algorithm (FMA) is adopted to accelerate the matrix vector product.

Skin-Effect-Incorporated Transient Simulation Using the Laguerre-FDTD Scheme

In this paper, skin effect is incorporated into the unconditionally stable Laguerre finite-difference time-domain method for transient simulation of multiscale structures. The skin effect is modeled by imposing the surface impedance boundary condition (SIBC) to the surface of the conductor, which significantly reduces the computational effort. Two methods of applying SIBC are proposed and discussed in detail. A procedure for transferring the time-domain convolution into Laguerre domain in recursive form is also discussed. In addition, a method for embedding and de-embedding port resistors for fast simulation and accurate extraction of frequency-domain response is proposed. The results have been validated using measurements, along with the stability analysis of the two implementation methods. Simulation results from the multiscale structures are presented and are compared with measurements which show good accuracy and improved computational efficiency.

Skin effect modeling of interconnects using the Laguerre-FDTD scheme

The Laguerre-FDTD scheme is unconditionally stable for transient electromagnetic simulation and is ideally suited for modeling multi-scale structures such as packaging and interconnects. In this work, skin effect is incorporated into Laguerre-FDTD to ensure fast simulation speed and high accuracy with less dense mesh applied. The skin effect is modeled by applying the surface impedance boundary condition (SIBC) on the interface of conductor and dielectric material. A method of transferring the time domain convolution term in SIBC formulation into Laguerre domain is proposed. Results from microstrip and TSV structures have been presented which show good calculation accuracy and efficiency of the SIBC incorporated Laguerre-FDTD method.

Transient Simulation of Multiscale Structures Using the Nonconformal Domain Decomposition Laguerre-FDTD Method

In this paper, a full-wave transient domain decomposition method is proposed for solving multiscale electromagnetic problems. The proposed scheme is based on the unconditionally stable Laguerre finite-difference time-domain (Laguerre-FDTD) method. The entire computational domain is partitioned into several subdomains with independent meshing according to the physical sizes of each subdomain. Standard Laguerre-FDTD method is used to form the interior field equations of each subdomain, whereas interface field equations are formed by applying the equivalency between finite-element method and FDTD method. In addition, field continuity is enforced through the use of two sets of Lagrange multipliers at the domain interface. Schur complement is implemented for extracting the interface problem and each domain can be evaluated independently. Numerical results for multiscale structures are presented to demonstrate the capability, accuracy, and efficiency of the proposed method.

An augmented electric field integral equation formulation for dielectrics and conductors at low frequencies

A full wave integral equation method for dielectrics and conductors is developed based on the augmented electric field integral equation. This method rigorously models dielectrics and conductors to solve circuit interconnects and packaging problems. By treating the two regions rigorously and solving both the internal and external problems with the augmented electric field integral equation, accurate results for dielectrics are obtained efficiently. The conductors, however, are modeled as highly lossy dielectrics. A novel integration technique is analyzed and adopted to capture the wave properties inside the conductors. A strategy of integrating this method with the mixed-form fast multipole algorithm is presented. With the fast algorithm, more complicated, multi-scale structures with a large number of unknowns can be solved. Some numerical results show the capability of this method to solve realistic circuit problems efficiently and accurately.

Memory efficient Laguerre-FDTD scheme for dispersive media

The unconditionally stable Laguerre-FDTD method is suitable for simulating 3-D structures with large time step. In this work, memory efficient Laguerre-FDTD scheme for dispersive materials is proposed to ensure accurate modeling and less memory consumption compared to standard procedures. The memory efficient scheme is realized by representing the Laguerre domain expression of electric susceptibility in a recursive manner. Formulations have been derived for both Debye and Lorentz media. Numerical results show that the proposed Laguerre-FDTD method exhibits significant peak memory usage reduction and equivalent calculation accuracy of dispersive material involved transient simulation.

An Integral Equation Modeling of Lossy Conductors With the Enhanced Augmented Electric Field Integral Equation

The formulation of the enhanced augmented electric field integral equation for dielectrics is generalized to conductor problems in this paper. The conductive region is simulated as a lossy dispersive medium using a full wave solver. In order to calculate the method of moments matrix elements in the conductive region accurately, we investigate the evaluations of the integrals of Green’s function in lossy media. After comparing with some other integration methods, we propose a new method to evaluate such integrals. This method turns out to improve the accuracy and efficiency. Moreover, the proposed formulation can be regarded as a generalized impedance boundary condition (IBC). This generalized IBC will become global if the skin depth is comparable to the size of the structures/details. The mixed-form fast multipole algorithm is employed for the simulations. Numerical examples of complex circuit structures are given to demonstrate the accuracy and capabilities of the proposed method.

A Cross-Layer Approach for Early-Stage Power Grid Design and Optimization

Power integrity has become increasingly important for sub-32nm designs. Many prior works have discussed power grid design and optimization in the post-layout stage, when design change is inevitably expensive and difficult. In contrast, during the early stage of a development cycle, designers have more flexibility to improve the design quality. However, there are several fundamental challenges at early stage when the design database is not complete, including extraction, modeling, and optimization. This article tackles these fundamental issues of early-stage power grid design from architecture to layout. The proposed methods have been silicon validated on 32nm on-market chips and successfully applied to a 22nm design for its early-stage power grid design. The findings from such practices reveal that, for sub-32nm chips, an intrinsic on-die capacitance and power gate scheme may have more significant impact than expected on power integrity, and needs to be well addressed at early stage.

Numerical conditional probability density function and its application in jitter analysis

Jitter is a critical factor to the performance of highspeed signal links. Jitter can be modeled as a random process. Both the probability density function (PDF) and the spectral characteristics of the jitter are important for evaluating the impact to the channel performance. The concept of numerical conditional probability density function (NCPDF) and a new statistical method called FastBER are proposed in this paper to accurately and efficiently perform the bit-error-rate (BER) analysis with taking into account both the PDF and the spectral characteristics of an arbitrary jitter sequence for arbitrarily low BER levels.