Efficient Implementation Algorithms for Homogenized Energy Models

The homogenized energy framework quantifying ferroelectric and ferromagnetic hysteresis is increasingly used for comprehensive material characterization and model-based control design. For operating regimes in which thermal relaxation mechanisms and stress-dependencies are negligible, existing algorithms are sufficiently efficient to permit device optimization and the potential for real-time control implementation. In this paper, we develop algorithms employing lookup tables which permit the high speed implementation of formulations which incorporate relaxation mechanisms and electromechanical coupling. Aspects of the algorithms are illustrated through comparison with experimental data. 1 Homogenized Energy Framework Ferroelectric and ferromagnetic materials are being considered as transducers for an increasing number of applications due to their broadband capabilities, large electromechanical and magnetomechanical coupling factors, and their dual capability for actuating and sensing. At low drive levels, the direct and converse electromechanical/magnetomechanical effects are approximately linear, and linear models and control designs can be employed. However, at the moderate to high drive levels where the unique transducer capabilities are manifested, the constitutive material properties are inherently nonlinear and hysteretic. Material characterization necessitates the development of models which accurately characterize constitutive nonlinearities and hysteresis whereas device optimization and real-time control implementation requires that the models be highly efficient to implement. While a number of frameworks have been developed for characterizing ferroelectric and ferromagnetic hysteresis, the competing requirements of accuracy and efficiency limit which models may be considered for both material characterization and real-time implementation. At the microscopic scale, the physical mechanisms which produce hysteresis in ferroelectric and ferromagnetic materials differ substantially since ferroelectricity is due to the ionic structure of materials and ferromagnetism results from interactions between magnetic moments and electron spins. However, at the domain or macroscopic scale, shared physical and energy properties permit the development of unified frameworks for characterizing hysteresis of compounds (see [14] for details regarding shared properties of ferroic materials at the various scales). In addition to the dielectric and magnetic hysteresis exhibited by the compounds, transducer models must characterize the thermal relaxation effects and stress-dependencies exhibited by ferroelectric and ferromagnetic materials. The former phenomenon is illustrated in Figure 1(a) with magnetic data collected from a steel rod [3]. Stress effects are illustrated in Figure 1(b) via PLZT data from [9]. As detailed in [14], several frameworks have been developed to characterize ferroelectric and ferromagnetic hysteresis for regimes in which relaxation mechanisms and stress-dependencies are negligible. These include ∗Email: [email protected]; Telephone: 919-854-2746 †Email: [email protected]; Telephone: 919-515-7552