Giorgio Casinovi

Passive TCF compensation in high Q silicon micromechanical resonators

This paper reports on passive temperature compensation techniques for high quality factor (Q) silicon microresonators based on engineering the geometry of the resonator and its material properties. A 105 MHz concave silicon bulk acoustic resonator (CBAR) fabricated on a boron-doped substrate with a resistivity of 10−3 Ω-cm manifests a linear temperature coefficient of frequency (TCF) of −6.3 ppm/°C while exhibiting a Q of 101,550 (fQ = 1.06×1013). The TCF is further reduced by engineering the material property via a wafer-level aluminum thermomigration process to −3.6 ppm/°C while maintaining an fQ of over 4×1012. Such high fQ products with low TCF values are being reported for the first time in silicon and are critical for successful insertion of these devices into low-power low-phase noise frequency references and high performance resonant sensors.

A macromodeling algorithm for analog circuits

A macromodel is an electrical network containing fewer devices and/or fewer nodes than the circuit it represents. A general-purpose algorithm for the generation of macromodels suitable for circuit simulation is presented. The algorithm is based exclusively on a comparison of the input-output behavior of the macromodel with that to the circuit to be modeled. Because no reliance on any particular properties of the circuit is made, the algorithm can be used to model a very wide class of circuits. Three examples are presented to demonstrate the algorithms performance, two of which were chosen specifically to test the algorithms ability to approximate nonlinearities in the circuits to be modeled. In all cases, the accuracy of the macromodels was shown to improve substantially using reasonable amounts of CPU time. >

Temperature-Stable Silicon Oxide (SilOx) Micromechanical Resonators

This paper presents a passive temperature compensation technique that can provide full cancellation of the linear temperature coefficient of frequency (TCF1) in silicon resonators. A uniformly distributed matrix of silicon dioxide pillars is embedded inside the silicon substrate to form a homogenous composite silicon oxide platform (SilOx) with nearly perfect temperature-compensated stiffness moduli. This composite platform enables the implementation of temperature-stable microresonators operating in any desired in- and out-of-plane resonance modes. Full compensation of TCF1 is achieved for extensional and shear modes of SilOx resonators resulting in a quadratic temperature characteristic with an overall frequency drift as low as 83 ppm over the industrial temperature range ( -40°C to 80°C). Besides a 40 times reduction in temperature-induced frequency drift in this range, SilOx resonators exhibit improved temperature stability of Q compared with their single crystal silicon counterparts.

A 145MHz low phase-noise capacitive silicon micromechanical oscillator

This paper reports on the implementation and characterization of a low phase-noise oscillator based on a very high quality factor (Q) 145MHz capacitive silicon micromechanical resonator. The utilized resonator is a silicon bulk acoustic resonator (SiBAR) operating in its first width-extensional mode with a maximum Qunloaded~74,000 that is specifically optimized for low motional impedance. The sustaining circuitry is a 3.6mW CMOS transimpedance amplifier (TIA) that uses common source topology with local shunt-shunt feedback. The measured phase-noise of the oscillator at 1kHz offset from the carrier is -111dBc/Hz with phase-noise floor reaching below -133dBc/Hz.

Accurate Transient Simulation of Interconnects Characterized by Band-Limited Data With Propagation Delay Enforcement in a Modified Nodal Analysis Framework

A numerical-convolution-based approach has been proposed for the accurate transient simulation of interconnects characterized by band-limited (b.l.) frequency-domain (f.d.) data and terminated by arbitrary equivalent circuits. Propagation delay is enforced in the transient results by obtaining causal impulse responses from b.l.f.d. data, extracting the propagation delays from them, and enforcing the delays in the causal impulse responses. Causal impulse responses are obtained through a new minimum- phase/all-pass decomposition of the frequency data. In this decomposition, a new form for the all-pass component has been proposed that preserves the sign of the original frequency response in the reconstructed response, unlike the prior approaches, leading to an accurate transient result. Arbitrary terminations are conveniently handled by integrating the numerical convolution in a modified nodal analysis (MNA) framework, a framework used by commercial circuit simulators, through a new transient simulation formulation. Numerical results demonstrating the accuracy and capability of the proposed procedure have been presented.

Multi-level simulation of large analog systems containing behavioral models

The trend towards integration of analog and digital components on the same chip is creating a need for simulators capable of handling high-level analog behavioral models. This paper presents a multi-level simulation algorithm that accepts analog behavioral models written in the C programming language. The portions of the circuit described at the behavioral level are treated as if they were individual devices with their own stamps. There are two main advantages to this approach: on one hand, it creates a seamless environment where there is no need to use different algorithms to simulate different parts of the same circuit. On the other hand, the proposed algorithm gets around the problem of having to come up with a reliable and efficient way to differentiate the behavioral equations, because the stamps for the elements described at the behavioral level are generated using Broydens method, which does not require evaluation of derivatives. Numerical results show that the algorithm offers a viable, efficient approach to multi-level analog simulation. >

Magnetic field near a concrete wall during a lightening stroke

The authors highlight the electromagnetic behavior of a lightning protection system with reference to a concrete wall making use of iron rods as vertical down conductors and horizontal equipotential conductors. The electrical model is obtained by dividing each rod into elementary cells, each represented by an equivalent symmetric pi network. During a lightning flash the current flowing through each cell and the magnetic field generated by it can be computed at any point in space at any time. The performance of this structure is compared with that of a conventional lightning protection system. >

Analytical Modeling and Numerical Simulation of Capacitive Silicon Bulk Acoustic Resonators

This paper introduces two newly developed models of capacitive silicon bulk acoustic resonators (SiBARs). The first model is analytical and is obtained from an approximate solution of the linear elastodynamics equations for the SiBAR geometry. The second is numerical and is based on finite-element, multi-physics simulation of both acoustic wave propagation in the resonator and electromechanical transduction in the capacitive gaps of the device. This latter model makes it possible to compute SiBAR performance parameters that cannot be obtained from the analytical model, e.g. the relationship between transduction area and insertion loss. Comparisons with measurements taken on a set of silicon resonators fabricated using electron-beam lithography show that both models can predict the resonant frequencies of SiBARs with a relative error smaller than 1%.

Electrostatic self-calibration of vibratory gyroscopes

This paper introduces a new approach to self-calibration of Coriolis-based vibratory gyroscopes that does not require the use of any additional moving parts or a calibration stage. Instead, the effect of the Coriolis force on the device is mimicked by the application of a rotating excitation to the device drive and sense modes. This calibration method is based on the theoretical analysis of an equivalent 2-DOF mass-spring model, which can be used to describe the behavior of a variety of MEMS gyroscopes. The method is validated both by the results of finite-element simulations and by experimental measurement.

Lamb Waves and Resonant Modes in Rectangular-Bar Silicon Resonators

This paper presents two newly developed models of capacitive silicon bulk acoustic resonators (SiBARs) characterized by a rectangular-bar geometry. The first model is derived from an approximate analytical solution of the linear elastodynamic equations for a parallelepiped made of an orthotropic material. This solution, which is recognized to represent a Lamb wave propagating across the width of the resonator, yields the frequencies and shapes of the resonance modes that typically govern the operation of SiBARs. The second model is numerical and is based on a finite-element multiphysics simulation of both acoustic wave propagation in the resonator and electromechanical transduction in the capacitive gaps of the device. It is especially useful in the computation of the SiBAR performance parameters, which cannot be obtained from the analytical model, e.g., the relationship between the transduction area and the insertion loss. Comparisons with the measurements taken on a set of silicon resonators fabricated using electron-beam lithography show that both models can predict the resonance frequencies of SiBARs with a relative error, which, in most cases, is significantly smaller than 1%.