Capacitive Micromotor

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• Capacitive micromotors are used as MEMS actuators

• In this tutorial we will explore a possible design

• The rotor and stator are made of polysilicon

• A pulsed voltage is applied on different cogs of the stator while the rotor is kept at electrical ground

• This produces a time-varying torque that drives the rotor

Introduction

V1 = V0*sin(ωt)V2 = V0*sin(ωt+2π/3)V3 = V0*sin(ωt+4π/3)

V = 0

V1

V1

V2

V2

V3

V3

• Physics interfaces used:– Electrostatics

• Computes the voltage distribution in the modeling region• Uses the solution to compute the torque acting on the rotor• A Global Equation is added to implement the equation of rotary motion

– Deformed Geometry• Allows the movement of computational mesh based on prescribed displacement• This helps in solving the electrostatics problem in an effectively modified geometry as the rotor

moves without really redrawing the geometry• The domains representing the stator and the air region around them are fixed• The rotor domain and the air region around it is rotated based on the angular displacement

obtained from the equation of rotary motion

Model Implementation

zzI

T

t

2

2 θ = angular displacementt = timeT = torqueIzz = area moment of inertia

Modeling steps

• The next few slides illustrate the key modeling steps

• The detailed steps are available in the file: capacitive_micromotor.mph

Start with the Model Wizard

Select Physics: Electrostatics & Deformed Geometry

Select Study

Geometry

The Form an Assembly option is needed here to create geometric discontinuity which is used later to allow the mesh around the rotor to slide against the mesh around the stator

Identity Pair• The Identity Pair is automatically created on building the geometry• It is used later to set up a boundary condition on these

geometrically discontinuous boundaries that allows the electric potential across these boundaries to be continuous while the mesh in the inner region is allowed to slide against the mesh on the outer region

Identity Pair boundaries

Assign Materials

• Air and Polysilicon are selected from the Built-In branch in the Material Browser

• The rotor and stator domains (shown in blue) are assigned to Polysilicon

• All other domains are assigned to Air

Create Parameters

Create 3 Square Wave Functions

Create Variables for Excitation Voltage

• The default square wave function varies the magnitude between -1 to +1• We want the magnitude of the pulse to vary between 0 to 1• This is achieved by adding of +1 to the expression wv1(t[1/s])

Add a Mass Properties Computation• Right-click on Component 1 >

Definitions to select Mass Properties

• This is used to compute the area moment of inertia of the rotor and generate a variable (mass1.Izz) which is used later

• Assign the rotor domain only• Set the density expression to

es.d*mat2.def.rho– es.d = out-of-plane thickness– mat2.def.rho = density of the 2nd

material listed under the Materials branch (i.e. Polysilicon)

Create 5 Explicit Selections

• This is used to group together certain boundaries that are used in the physics and mesh settings later

• The details of the settings can be seen in the model file• Ground selection contains the boundaries of the rotor domain• V1, V2 and V3 selections contain the boundaries of the

respective stator domains to which we apply voltages V1, V2 and V3 (as defined in the Variables branch) respectively

• Destination selection contains the Destination boundaries of the Identity Pair

Add Infinite Element Domain

• Assign this to the outer layers of the air domain

• Accounts for electrostatic energy stored in an infinitely extended region of air

• More accurate computation of torque

Electrostatics

• Deselect the stator domains as each of them will be under a different isopotential condition dictated by the voltage on their boundaries

• Assign the correct out-of-plane thickness which is needed to compute the correct magnitude of the torque acting on the rotor

Assigning Ground and Voltages

• These are the initial voltages on the different cogs of the stator at time t = 0

• This information is used to solve a stationary study, the solution of which provides the initial condition (a consistent spatially varying potential distribution) for the subsequent time-dependent study

Ground V1

V2 V3

Duplicate the 3 Electric Potential branches

• Use Ctrl-click to select the 3 Electric Potential branches• Right-click and select Duplicate• Specify the voltages on the 3 new branches as shown below• Rename the branches so that we know which boundary conditions

should be used in the Time-dependent Study

Setting Continuity of Electric Potential

Adding a Force Calculation

• This computes the electrostatic forces and torques acting on the rotor• Note that the default setting for Torque axis and Torque rotation point is

appropriate for this model but may need to be changed based on the geometry and physics of the problem

Adding a Global Equation

• Check Advanced Physics Options• This activates the Global button in

the Physics ribbon• Browse to add a Global Equation

under Electrostatics

Setting up the Equation of Rotary Motion

• u = angular displacement of rotor• utt = angular acceleration• es.Tz_rotor = out-of-plane torque• mass1.Izz = area moment of inertia

Recall equation of motion:

In COMSOL:utt – es.Tz_rotor/mass1.Izz

zzI

T

t

2

2

Deformed Geometry

• These expressions are used to make the inner region undergo rigid body rotation based on the computed angular displacement

• Xg and Yg denote the coordinates of the Geometry frame that is associated with the Deformed Geometry interface

Mesh sequence

• Use a Mapped mesh on the Infinite Element Domains– Specify a distribution of 5 elements through

the width

• Use a Free Triangular mesh on all other domains– Specify a maximum mesh element size of 2 μm

on the Destination boundaries of the Identity Pair to resolve the continuity in the solution better across these boundaries

Add a Time Dependent Study Step

• We will solve a two-step analysis• Stationary step only solves the Electrostatics

problem on the original geometry using constant voltages at different regions of the stator

• Time Dependent step uses the solution of the Stationary step as an Initial Value for the electric potential distribution and solves for Electrostatics with time-varying excitation voltages, the Global ODE for angular displacement and the Deformed Geometry

Set up the Stationary Step• Click on Step 1: Stationary• Cross out Deformed Geometry by

clicking on the green check next to it so that it turns to a blue cross

• Check the Modify physics tree and variables for study step

• Use ctrl-click to select the branches shown with arrows and click on the blue Disable button below the list– Do not disable the Continuity branch

• Click on the Deformed Geometry branch and click on the blue Disable button below the list

Set up the Time Dependent Step

• Click on Step 2: Time Dependent

• Check the Modify physics tree and variables for study step

• Use ctrl-click to select the branches shown with arrows and click on the blue Disable button below the list

Solvers

• Generate the default solver configuration• Browse to Study 1 > Solver Configurations >

Solver 1 > Time-Dependent Solver 1 > Fully Coupled 1

• In the settings window, expand the Method and Termination section and set the Jacobian update to be done Once per time step– This provides a more robust solver setting especially

when the physics set up involves using the Deformed Geometry interface

Create a Probe to track the Angular Displacement of the Rotor

• This will allow us to track the variation in angular displacement with time while solving the model

• You are now ready to Compute

Probe Plot of Angular Displacement

Electric Potential Distribution• Enable full-screen to view movie

Excitation Voltage Profiles

Torque on Rotor

Summary• This tutorial showed how to model a capacitive micromotor in

2D time-dependent model• Key modeling steps:

– Solve for an electrostatic problem to find spatial distribution of voltage around rotor and stator

– Use this information to find the torque acting on the rotor– Find the angular displacement of the rotor by solving the equation of motion

that uses the computed torque and the moment of inertia of the rotor– Use this information to rotate the mesh using the Deformed Geometry

interface

• Important results– Electric potential distribution– Angular displacement of rotor– Torque acting on rotor