Research/Multi-physics Analysis for MEMS based on Electrical Circuit Simulator Spice

2019-02-18 (月) 00:18:11 (243d)

LTspiceを用いたCMOS−MEMSのためのマルチフィジクス解析手法に関する研究
Multi-physics Analysis for MEMS based on Electrical Circuit Simulator Spice

MEMS分野は電気系、機械系、化学系、流体系、光学系など、さまざまな領域が融合した分野であるため、複数の物理現象を同時に取り扱うマルチフィジクス的な統合解析・設計手法が必要です。しかしながら、従来のMEMS解析ソフトでは、マイクロ機械構造の3次元的な動作の解析に主眼が置かれておりました。このため、MEMSアクチュエータ・センサを電気回路と組み合わせたときのシステム全体の挙動を解析する用途には、案外、複雑なCAD操作が必要でした。そこで本研究では、機械系の運動方程式とアクチュエータ出力を等価回路モデル化して、電気回路シミュレータ上で解析する簡便な手法を考案しました。

Multi-physics simulation is becoming an indispensable technique for integrated MEMS (micro electromechanical systems) to comprehend the system level behavior including electronics, mechanics, and/or other domain of physics such as optics, acoustics, and fluidics, depending upon the type of micro sensors and actuators (1). We have newly developed a single platform simulation technique for MEMS by using a compact model solver for the mechanical equation-of-motion (EOM) implemented as an analog computation circuit on an electrical circuit simulator such as Qucs (2), LTspice (3), and Cadence Virtuoso. In this paper, we review the conventional methods to develop an equivalent circuit model for MEMS, and then report our own simple and useful technique to create a compact model for electrostatic micro actuators and sensors by translating the algebraic analytical model into an equivalent circuit model based on the non-linear independent current source.

はじめに
Introduction

Fig1.png

従来の解析手法では、たとえば垂直櫛歯型アクチュエータのフォトマスク(2次元)をもとに3次元メッシュモデルを構築して、機械的な応力と歪の関係を計算するとともに、同じモデルを静電解析にも利用して、変位と静電容量の関係を計算します。さらに、変位-静電容量の関係から静電引力(dC/dx)の関係を導出します。これらの結果をデータベース化して統合設計ツールに手渡すことで、電圧と機械歪みの関係を計算する方法が一般的でした。このため、マスク上の一部を修正すれば、一連の解析を最初からやり直す必要がありました。

The figure on the left illustrates the conventional procedure for MEMS actuator analysis based on the finite element method (FEM). One would usually use the photomasks to construct a three dimensional meshed model for a micro actuator by considering the material and processed used for microfabrication. In the mechanical path shown in the left-hand side of the chart, we calculate the mechanical characteristics of the device, such as displacement as a function of applied force, and prepare a look-up table. In the path on the right-hand side, the identical meshed model is put into an electrical field solver to calculate the electrical capacitance between the elements as a function of mechanical displacement. The data is then converted into another look-up table of force and displacement. These two sets of data are cross-referenced in the electromechanical solver, which is usually referred to as a co-solver, to numerically calculate the mechanical displacement as a function of applied voltage. A drawback of this method is, however, that one may need to repeat the whole procedure from the beginning every time the MEMS device dimensions are modified, and that it takes large amount of computation power of PC to process the mechanical and electrical matrix data.

等価回路モデル&brEquivalent Circuit Model

Fig2.png

一方、本研究の手法では、静電アクチュエータやサスペンションなどの部品の入出力関係を解析的にモデル化して、それを数式で表します。その上で、回路シミュレータの非線形従属電流源(数式で電流値を記述する)を用いて、力と変位を電流や電圧で表現します。これらを電気回路シミュレータの上で表現して、あたかもアナログ計算機を走らせるようにして、機械系の運動方程式を解析します。

Different from the conventional approach, we used a very simple method to simulate the behavior of a MEMS actuator or sensor by using an electrical circuit simulator a platform for multi-physics analysis. We parameterize a micro actuator with its dimensions to express the output force or torque by using a lumped equivalent circuit model. We also have developed a solver for the mechanical equation of motion in the same manner as an analog computing. As a result, all the elements of micro actuator are represented by lumped parametric models including such as a suspension, actuator plates, a moving mass, and a mechanical anchor.

Fig3.png

本研究の手法を用いて平行平板型静電アクチュエータの変位-電圧関係を解析した結果を示します。このアクチュエータには、初期ギャップの1/3だけ変位した時点で変位が跳躍する「静電プルイン」と呼ばれる現象が知られています。本研究の解析では、この様子が再現できており、紙と鉛筆を用いた解析的な計算結果(図の◯印)とよく一致しています。しかも、本研究の手法では、プルイン後の変位の固着の様子と、電圧降下後の変位リリースのタイミングまで計算できるという特徴があり、静電アクチュエータ固有の電圧-変位ヒステリシス・ループを忠実に再現できています。

Figure on left compares the known analytical model (dots) and the numerical simulation results (solid curve). With increasing the drive voltage, the movable plate is attracted toward the fixed one; when it moves 1/3 of the initial gap g, the plate is found to trip to the stopper position. This phenomenon is known as the electrostatic pull-in, where the electrostatic attractive force exceeds the mechanical restoring force. Unlike the conventional analytical model, the developed one is found to reproduce the entire hysteresis loop including the electrostatic pull-in and the following release process. Pull-in effect is widely used in electrostatic actuators such as digital mirror devices, grating light values, and RF-MEMS switches in presence of electronic circuitry, and hence it is an advantage of using the electrical circuit simulator as a platform for the multi-physics simulation of MEMS. The identical simulation model can also be used for harmonic analysis to predict the frequency response of the amplitude of mechanical oscillation at given DC bias voltages.

解析例
Simulation Example

Fig4.png

また、本研究の手法はベースが電気回路シミュレータなので、機械系と電気系の統合解析は当然可能です。図では、静電駆動アクチュエータをシリコン共振子として扱って、コルピッツ共振回路を用いて発振させる様子を示しています。

It is naturally possible to simulate both electrical and mechanical system at the same time because we use the electrical circuit simulator to implement the equation-of-motion solver. The figure on the left shows an example, where we use the electrostatic micro actuator as a silicon resonator inserted into a Colpitts oscillation circuit.

まとめ&brConclusion;

Table1.png

本研究の手法では、サスペンションや静電駆動電極、運動方程式(質量)を集中定数として扱っていますので、短時間での計算が可能です。安定した発振を持続するパラメタ条件が分かったら、そのパラメタを実現するための機械的構造を、従来の3次元CADを用いて設計します。これにより、最初から3次元CADを用いるよりも、効率よくシステム全体のパラメタ最適化することが可能です。

In conclusion, we have used a common electrical circuit simulator to build a user-friendly analysis environment for MEMS multi-physics simulation. Our approach takes a lumped parametric model to describe an electromechanical component and interpret it as an electrical equivalent circuit. A key to understand the multi-physics capability is in the kernel co-solver for the mechanical equation-of-motion that has been implemented as a 2nd order integration circuit with feed-back loop with a suspension module and an electrostatic actuator module. Our approach is more straightforward because no labor is needed to extract parameter for the SPICE net-list from the mechanical network but what-you-see in the simulation diagram is a direct translation from what-you-have in an actual MEMS design.

紹介ポスター&Poster