Extended and Generic Higher-Order Elements for MEMS Modeling

Abstract

State-dependent resistors, capacitors, and inductors are a common part of many smart engineering solutions, e.g., in MEMS (Micro-Electro-Mechanical Systems) sensors and actuators, Micro/NanoMachines, or biomimetic systems. These memory elements are today modeled as generic and extended memristors (MR), memcapacitors (MC), and meminductors (ML), which are more general versions of classical MR, MC, and ML from the infinite set of the fundamental elements of electrical engineering, known as Higher-Order Elements (HOEs). It turns out that models of many complex phenomena in MEMS cannot be constructed only from classical and state-dependent elements such as R, L, and C, but that other HOEs with generalized behavior should also be used. Thus, in this paper, generic and extended versions of HOEs are introduced, overcoming the existing limitation to MR, MC, and ML elements. The relevant circuit theorems are formulated, which generalize the well-known theorems of classical memory elements, and their application to model complex processes of various physical natures in MEMS is shown.

Extended Higher-Order Elements with Frequency-Doubled Parameters: The Hysteresis Loops Are Always of Type II

“…− , .., P (i−1) + = (−1) i−1 P (i−1) − (10) where the indices + and − denote the correspondence between the positive and negative arms of the PHL. It follows from (10) that for the terms (7) of sequence (6) with indices i > 1, it holds:…”

“…Without loss of generality, this element can be considered an extended memristor, where y is the voltage and u is the current. Let the amplitude and frequency of the current sinusoidal excitation (10) be I = 0.1 A and f = ω/(2π) = 10 Hz. In the three subsequent simulations, the constants P 0 =20 and k 2 =100 remain fixed.…”

“…This concept allows treating the already known elements of resistor, capacitor, inductor, memristor, memcapacitor, and meminductor uniformly as HOEs (0,0), (0,−1), (−1,0), (−1,−1), (−1,−2), and (−2,−1). The element defined by (1) and (2), whose constitutive variables u and y can be the pairs ( v ( α ) , i ( β ) ), is called the Extended Higher-Order Element (Extended HOE) [ 10 ]. Generic elements differ from extended elements in the state-dependent parameter P ( x ), which does not depend on the excitation u [ 65 ].…”

“…Similarly, if the state variable x ( t ) oscillates at a frequency that is an even multiple of the excitation frequency, then the native loop type is type II, i.e., non-crossing type. In [ 10 ], these findings were generalized to extended HOEs.…”

“…Similar manifestations of hysteresis can also be provided by subsystems of non-electrical nature, especially the mechanical parts of MEMS, or they can be the results of interactions between subsystems of different nature. Such mechatronic mem-systems can be effectively studied using the generalization of so-called Chua’s table and predictive modeling methods [ 8 ] to the non-electrical domain [ 9 ], as well as the generalization of memristive, memcapacitive, and meminductive systems to so-called extended and generic Higher-Order Elements (HOEs) [ 10 ].…”