1. Capacitive Micromotor
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2. Introduction 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
V = 0 V1 V2 V3
V1 = V0*sin(ωt)
V2 = V0*sin(ωt+2π/3)
V3 = V0*sin(ωt+4π/3)

3. Model Implementation 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
θ = angular displacement
t = time
T = torque
Izz = area moment of inertia

4. Modeling steps The next few slides illustrate the key modeling steps
The detailed steps are available in the file: capacitive_micromotor.mph

5. Start with the Model Wizard

6. Select Physics: Electrostatics & Deformed Geometry

7. Select Study

8. 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

9. Identity Pair Identity Pair boundaries
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

10. 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

11. Create Parameters

12. Create 3 Square Wave Functions

13. 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])

14. 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)

15. 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

16. 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

17. 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

18. 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 V2 V3

19. 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

20. Setting Continuity of Electric Potential

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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

31. Probe Plot of Angular Displacement

32. Electric Potential Distribution
Enable full-screen to view movie

33. Excitation Voltage Profiles

34. Torque on Rotor

35. 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