Thin Film Bulk Acoustic Resonators (FBARs) are widely used in the design of modern radio frequency components including duplexers, filters, and oscillators. The overall goal of this project is to incorporate the performance parameters of these resonators into the design flow of the overall system. As a first step, the frequency response of the fabricated devices is measured. Traditionally, an equivalent circuit is then built based on least squares fitting of the frequency response of a simple RLC network to the measured data ^[1] . Such a technique is fairly simple, and the resulting equivalent model does capture important performance parameters, such as quality factor and resonant frequency. However, this technique cannot capture spurious resonances and other second order effects, which quite often play a significant role in the overall performance of the device.

In this work, we are developing tools that will automatically generate accurate, compact, and passive dynamical models for FBARs. Given measured transfer function samples, we identify a rational transfer function model that minimizes the mismatch at the given frequencies. These dynamical models can be interfaced with commercial circuit simulators for time domain simulations of a larger interconnected system. To guarantee the stability of the overall simulation, we ensure the passivity of our generated models by enforcing semidefinite constraints during the fitting process as proposed in ^[2] . Figure 1 shows the 3D layout of an FBAR. Numerical results are presented for resonators configured to constitute a bandpass frequency response. Figure 2 compares the output of our identified models with the given measured data.

1. R. C. Ruby, P. Bradley, Y. Oshmyansky, A. Chien, and J.D. Larson, III, “Thin film bulk wave acoustic resonators (FBAR) for wireless applications,” Ultrasonics Symposium, 2001 IEEE, vol.1, no., pp. 813-821.
2. Z. Mahmood and L. Daniel, “Circuit synthesizable guaranteed passive modeling for multiport structures,”in Proc. Behavioral Modeling and Simulation Conference (BMAS), Sept. 2010.

During recent years, researchers of the Electronic Design Automation community have made a great effort to develop new techniques for automatically generating accurate compact models of NON-LINEAR system blocks. The majority of the existing methods for creating stable reduced models of nonlinear systems, such as ^[1] , require knowledge of the internal structure of the system, as well as access to the exact model formulation for the original system.  Unfortunately, this information may not be easily available if a designer is using a commercial design tool, or may not even exist if the system to be modeled is a physical fabricated device.

As an alternative approach to nonlinear model reduction, we have proposed a system-identification procedure.  This procedure requires only data available from simulation or measurement of the original system, such as input-output data pairs.  By enforcing incremental stability, as shown in ^[2] , it is possible to formulate a semi-definite optimization problem whose solution is a stable nonlinear model that optimally matches the given data pairs from the original system.  In addition, the proposed optimization formulation, explained in detail in ^[3] , allows users to specify completely the complexity of the identified reduced model through the choice of both model order and nonlinear function complexity.

Applications for the proposed modeling technique include analog circuit building blocks such as operational amplifiers and power amplifiers, MEMS devices, and individual circuit elements such as transistors.  The resulting compact models may then be used in a higher-level design optimization process of a larger system.   One such example of an analog circuit block is the low-noise amplifier shown in Figure 1; it contains both nonlinear and parasitic elements.  For this example, input-output training data was generated from a commercial circuit-simulator and used to identify a compact nonlinear model.  The output responses of the original system and the identified model are compared in Figure 2.

1. B. Bond and L. Daniel, “Stabilizing schemes for piecewise-linear reduced order models via projection and weighting functions,” in Proc. IEEE Conference on Computer-Aided Design, San Jose, CA, Nov. 2007, pp. 860-867.
2. A. Megretski, “Convex optimization in robust identification of nonlinear feedback,” in Proc. IEEE Conference on Decision and Control, Cancun, Mexico, Dec. 2008, pp. 1370-1374.
3. B. Bond, Z. Mahmood, Y. Li, R. Sredojevic, A. Megretski, V. Stojanovic, Y. Avniel, and L. Daniel, “Compact Modeling of Nonlinear Analog Circuits Using System Identification via Semidefinite Programming and Incremental Stability Certification,” Computer-Aided Design of Integrated Circuits and Systems, IEEE Transactions on , vol.29, no.8, pp.1149-1162, Aug. 2010.