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Spectral domain statistical analysis of RF circuits, combining a circuit simulator, which models the circuit part and a full-wave field solver, which models the passive elements, is presented in this paper. This paper first illustrates the importance of the knowledge of correlation information in accurately modeling the probability density functions (PDFs) of eventual objective functions using a simple transmission line paradigm. Next, this paper looks at the statistical study of on-chip RF passives using the spiral inductor as an example. It is shown that larger process variations necessitate modeling by means of a quadratic response surface to preserve accuracy. This results in nonindependent non-Gaussian nonclosed-form PDFs for the equivalent-circuit parameters of the passives. This paper then proposes a hierarchical technique to perform statistical analysis of RF circuits based on y-parameter representation for the circuit and the passive element parts. The proposed technique obviates the need for optimization steps to derive the equivalent-circuit parameters for the electromagnetic objects and the need to compute the correlation matrix between the circuit equivalent elements, while maintaining accuracy. The proposed approach is illustrated for the statistical analysis of an RF amplifier and its differential version operating at 15.78 GHz. PDFs of various quantities of interest are derived and yield measures are computed.

The increasing need for more computing power, the integration of multiple functionalities and the trends towards miniaturization of electronic systems have all contributed to rapid advancement in semiconductor and associated technologies. Higher frequencies of operation together with non-negligible coupling arising from form-factor size constraints, and inability to ignore parasitics imply that full-wave EM (Electromagnetic) simulation is an essential part of electronic system design.

Full-wave simulation and design of interconnect systems is becoming increasingly important in view of the continuing trend towards miniaturization and dense integration of components in electronic systems. While full-wave simulation methods have advanced rapidly over the last decade, there is a growing need for superior automated design tools. In this paper we first motivate the need for better design space exploration in high dimensions and outline the key ingredients that are imperative for the development of a successful simulation-based automated design tool. In this direction we describe the application of Orthogonal Arrays as an efficient sampling scheme and the Gaussian Process Regression as a promising model generation scheme for high dimensional design space exploration. We also describe an adaptive strategy for the model refinement and illustrate the same with a full-wave multi-conductor transmission line example. Finally we conclude by outlining the open problems and research challenges in this direction.

This paper proposes a methodology and framework for rapid active-passive co-synthesis of radio frequency circuits. The presented approach leverages advances in accelerated three-dimensional electromagnetic simulation technology to construct Maxwell-accurate parametric macromodels of on-chip passives, in particular spiral inductors. These macromodels can be used in the context of SPICE-level synthesis, thereby enabling concurrent sizing of passive and active components of radio frequency circuits. Moreover, macromodels obviate the need for topology exploration and parametric RLC model generation via optimization for on-chip passives. The co-synthesis framework is enabled by nonlinear, hyper-dimensional regression for macromodel generation and a simulated annealing based optimization scheme. As examples, and to demonstrate the efficacy of the proposed approach, two standard low-noise amplifier topologies are synthesized with tight performance constraints by co-optimization of circuit parameters and inductor geometries.

Synthesis of multi-GigaHertz radio frequency circuits brings together difficult challenges related to simulation, extraction and multidimensional space search. The standard approach of mapping all electromagnetic parasitics into parametric RLC models prior to synthesis is extremely restrictive especially when broadband and full-wave models with high accuracy are needed. In the presented approach, a two-stage macromodel that creates broadband, accurate parametric representations of passives, in particular spiral inductors, is developed. The broadband nature is captured through the Vector Fitting algorithm. The macromodels are implemented via efficient nonlinear, multidimensional regression using Relevance Vector Machine, and are coupled into circuit simulators through admittance parameters. Subsequently, optimization on both active and passive parameters are carried out simultaneously, thereby bypassing the ad hoc nature of two stage (actives and passives) approximate optimization. Two standard low-noise amplifier topologies are synthesized with tight performance constraints at center frequencies 5, 10 and 12 GigaHertz in order to demonstrate the frequency scalability of the methodology.

A parametric model of a generic differential ten-layer microprocessor package line with two motherboard layers has been developed. A dimensionality reduction scheme and a reusable, multi-dimensional look-up table precede the global optimization phase which is facilitated by a smooth interpolation scheme based on splines. An accelerated boundary element based full-wave electromagnetic solver has been used to construct a look-up table of S-parameters for a number of designs. The second phase features a custom local optimizer incorporating all the variables without any dimension reduction. This methodology has been applied to automated synthesis of the differential package line resulting in a significant improvement of the return loss performance.

A geometry based parametric model of a differential high-speed line traversing a ten-layer microprocessor package is developed. This model is used to undertake a detailed study of the effect of the various geometrical parameters on the return loss performance of such a package. The forward problem is solved using a fast, full-wave electromagnetic (EM) solver. The effect of various types of routing in the intermediate layers is examined closely. A sensitivity analysis based on parametric sweeps is carried out to identify the key variables. A hierarchical response surface based optimization is carried out to arrive at an optimum structure. A global optimizer based on simulated annealing is harnessed to find the optimum of non-linear and, in general, non-convex functions. The optimized structure exhibits excellent return loss characteristics translating into higher channel bandwidth.

Modeling process variations in circuits is becoming necessary as the technology continues to shrink in size. When this is coupled with increasing frequencies of operation of circuits in communication systems (>10 GHz), the use of full-wave field solvers is necessary to model the on-chip passive components such as spiral inductors and transformers. Thus it becomes necessary to couple full-wave field solvers and SPICE-like simulators for accurate statistical analysis of RF circuits. The present paper reviews the issues associated with using circuit models for electromagnetically modeled passive elements in conjunction with SPICE for statistical analysis. It then proposes a method which circumvents the outlined limitations.

This paper presents a statistical study of the various parameters of interest of an on- chip spiral inductor in the 10 GHz range using the PMCHWT formulation [1]. Variability in the conductivity of the CMOS grade Silicon substrate and trace metal widths are assumed. We show that EM analysis of these inductors can accurately help in understanding the impact of process variations especially at high frequencies. The use of the PMCHWT formulation also permits analysis of the electrical properties of the substrate that are particularly important at higher clock and carrier frequencies. To enable fast extraction of statistics, a simple 2D Lagrange interpolation technique is used to approximate the surfaces of the quantities of interest and then a Montecarlo operation is carried out on the interpolated 2D polynomial. We extract the probability density functions (PDFs) of the effective resistance (R), the inductance (L) and the quality factor (Q) of a spiral on a lossy Silicon substrate.

Micromachined antennas are recieving great interest as carrier frequencies move higher into the frequency spectrum due to their superior performance and amenability for integration with active devices. However their design is cumbersome owing to the complexity of the structure. To overcome this, in this paper, an iterative procedure is suggested to facilitate fast design of micromachined patch antennas based on a simulation study. A microstrip line on a micromachined Silicon substrate is simulated in a full wave simulator by solving for the ports only. From the obtained propagation constant, the effective dilectric constant for the micromachined substrate is estimated. The process is repeated for a number of values of the width of the microstrip and a plot is made for the variation of the effective dielectric constant with the microstrip width. Then an iterative method in combination with the extrapolated permittivity which includes the effect of cavity extensions in all the directions, is used to obtain the width and the corresponding effective dielectric constant. This method has been verified to be quite accurate by comparison with full wave simulations and hence it can function as a good starting point for designers to design micromachined antennas.