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```Electron Focusing Through a Magnetic Lens
John Dunec, Ph.D.
VP of Sales, NW USA
© 2012 COMSOL. COMSOL and COMSOL Multiphysics are registered trademarks of COMSOL AB. Capture the Concept,
COMSOL Desktop, and LiveLink are trademarks of COMSOL AB. Other product or brand names are trademarks or registered
Welcome to the Lunch-Time Tutorials!
• Solve One Problem Using COMSOL Multiphysics
• This Tutorial: Electron Focusing Through a Magnetic Lens
• 30-35 minutes duration
• Short Q&A at end
• Archived at:
www.comsol.com/webinars
Upcoming Tutorials:
• Magnetophoretic cell separation
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Presentation, Step-by-Steps, and COMSOL model available on request
Individual Physics you Learned in School
Heat in a rod, …
Stress in a wrench
• Individual equation sets … Applied to simple,
(and sometimes not-so-simple) single-physics problems
In Reality – Multiple Sets of Physics Interact
• Typically bi-directional nonlinear coupling
between multiple physical processes
Multiphysics: Multiple Interacting Phenomena
Could be simple:
• Heat
• Convected by Flow
Could be complex:
• Flow
– Navier-Stokes in tubes
– Porous flow in plug
• Mass Transport
– Three chemicals: A,B,C
– Reacting: A+2B → C
• Heat Transfer
– Exothermic reaction
– Reaction rate temperature dependant
B
COMSOL Multiphysics Solves These!
• Multiphysics – Everything can link to everything.
• Flexible – You can model just about anything.
• Usable – You can keep your sanity doing it.
• Extensible – If its not specifically there…add it!
Trusted by 80,000+ Users Worldwide
Anywhere you can type a number …
you can type an equation
• Or an interpolation function …
• And it can depend on anything known in your problem
• Example: Concentration-dependant viscosity:
Low concentration,
High velocity
  0.001 1 2c
2

High concentration,
Low velocity
Electron Focusing Through a Magnetic Lens
Key Elements
• Simulating a magnetic field from
an electric coil
• Setting up a multi-turn coil in
COMSOL
• Using Particle Tracing to plot
trajectories of electrons
responding to magnetic field
Two Physics: Magnetics & Particle Tracing
Magnetics
• Particles respond to Magnetic Forces
Particle Tracing
COMSOL Products Used – This Tutorial
• COMSOL Multiphysics, AC/DC Module
• Along with the Particle Tracing Module
supported by Autodesk, Inc. and/or any of its affiliates and/or subsidiaries. CATIA® is a registered trademark of Dassault Systèmes S.A. or its affiliates or subsidiaries. SolidWorks®
is a registered trademark of Dassault Systèmes SolidWorks Corporation or its parent, affiliates, or subsidiaries. Creo™ is a trademark and Pro/ENGINEER® is a registered trademark
of Parametric Technology Corporation or its subsidiaries in the U.S and/or in other countries. MATLAB® is a registered trademark of The MathWorks, Inc.
First: Setup and Solve Magnetics Problem
• Choose predefined multiphysics
• Import geometry sequence
• Define materials (Copper and Air)
• Set up Coil
• Mesh
• Solve
Magnetic Field
Surrounding Multi-turn
Electric Coil
• Air domain with
electrons
• Copper mechanical
structure
Copper
Coil
Air
Copper
• Multi-turn
electric coil
• Electron “Inlet” at
bottom of bottom
copper annulus
Copper
Electron
“Inlet”
Magnetic Insulation on all Outer Boundaries
Geometry
1st Solve for Magnetic Field
• Solve based on Magnetic Potential
  H  Je
B   A
• Ampere’s Law relates H and B
B  0 r H
• Applied Volumetric Current from a Multi-turn Coil
Je 
N I coil
Awire crossarea
ecoil
Modeling Nuts & Bolts: Coil Ref Circle
We Need to Define a “Reference Edge”
• To Trace the Path of the Current and …
• To Determine the Average Coil Length
Draw a mid-coil circle on
top of actual coil domain
Let’s do this in COMSOL …
Solution Should be Done
• 175,000 Degrees of Freedom
• 25 seconds on my desktop
Next Add Particle Tracing for Electrons
• Particle Tracing, Transient
Define Particle Physics
• Define the Particle Properties
• Create an Inlet (500 electrons, velocity = 1.33e7)
Set up Transient Study
• Time Stepping
• Link Transient Particle Tracing to Magnetic Stationary Results
Lorenz Force on the Electrons
• Newton’s Law – recast to change in momentum
d
F  ma  mv
dt
• The force will be the Lorentz force of a particle in a
magnetic field – which changes the momentum
d
mv  qv  B
dt
• v is particle velocity vector, q is the particle charge,
m is the particle mass
Magnetic Lens
• Solve for Magnetic Field from
simple Coil
• Apply Magnetic Force to electron
particles entering from below
• Vector cross product with B-field
causes particles to both spiral &
focus
d
mv  qv  B
dt
B-Field Vectors
Let’s do this in COMSOL …
Solution Should be Done
• 1,500 Degrees of Freedom
• 9 seconds on my desktop
Tutorial Review
First: Setup and Solve Magnetics Problem
• Import geometry sequence
• Define materials (Copper and Air)
• Set up Coil
• Mesh & Solve
• Setup Particle Properties
• Set up 500 electrons at inlet
• Set up time-steps, link to magnetics
• Solve again
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TM
Capture the Concept
Step-by-step
Tutorial
Start by Solving for Magnetic Field
• Choose File > New , 3D
• Select “AC/DC” >
“Magnetic Fields”
• Choose “Stationary”
Units – Millimeters
• Highlight “Geometry1”
• Set units to “mm”
Set up Geometry – Electromagnet Plane
• Right click on Geometry
• Choose “Workplane”
• Select “Quick plane” and
“yz-plane”
• Choose the “Show Workplane”
button
• Air domain with
electrons
• Copper mechanical
structure
Copper
Coil
Air
Copper
• Multi-turn
electric coil
• Electron “Inlet” at
bottom of bottom
copper annulus
Copper
Electron
“Inlet”
Magnetic Insulation on all Outer Boundaries
Geometry
Focusing Geometry: Bottom Copper Ring
• Right click on Plane Geometry
(under Workplane 1)
• Choose: Rectangle
• Width: 8
• Height: 2.5
• Position: Corner
• X: 2
• Y: -7.5
• Build, Zoom Extents
Focusing Geometry: Middle Copper Ring
• Right click on Plane Geometry
(under Workplane 1)
• Choose: Rectangle
• Width: 7
• Height: 2.5
• Position: Corner
• X: 3
• Y: -2.5
• Build, Zoom Extents
Focusing Geometry: Top Copper Ring
• Right click on Plane Geometry
(under Workplane 1)
• Choose: Transformations
> Copy
• Select the top rectangle
• Select “Keep input objects”
•
•
•
•
Displacement:
xw: 0
yw: 5
Build, Zoom Extents
Focusing Geometry: Inner Copper Coil
• Right click on Plane Geometry
(under Workplane 1)
• Choose: Rectangle
• Width: 4
• Height: 2.5
• Position: Corner
• X: 6
• Y: 0
• Build, Zoom Extents
Revolve these into 3D
• Rt Click on “Work Plane 1”
• Choose “Revolve”
• Leave End Angle as “360”
• Build, zoom extents
Draw the Electron Inlet Surface
•
•
•
•
Rt Click on “Geometry 1”
Choose “Workplane”
Select “Face Parallel”
Pick the bottom face of the copper
•
•
•
•
•
•
•
Choose the “Show Workplane” Icon
Choose “Zoom Extents”
Pick the Draw centered circle” button
Draw a circle as shown
(Line up with the inner circle)
Build all
Highlight “Geometry” > Build all
Geometry: Surrounding Air
• Rt click on “Geometry 1”
• Select “Cylinder”
• Height: 50
• Position: z = -20
(Negative!)
Materials
• Rt Click on “Materials”
• Choose “Material Browser”
•
•
•
•
Expand “Built-in”
Rt click on “Copper”
Make sure “All domains” is selected
•
•
•
•
Pick “Material Browser” tab
Rt click on “Air”
Select the outer Air Domain
Specify Coil Domain
• Rt Click on “Magnetic Fields”
• Add “Multi turn Coil Domain”
• Select the 2nd Annulus from top
• Change “Coil Type” to “Circular”
• Enter Number of Turns as “1000”
• Coil Current as “0.32[A]”
Reference Circle for Coil
We Need to Define a “Reference Edge”
• To Trace the Path of the Current and …
• To Determine the Average Coil Length
Draw a Mid-Coil Circle
• Rt Click on Geometry 1
• Choose “Workplane”
• “Quick Plane” “x-y” z-coord: “2.5”
• Choose “Show Workplane” > Zoom Extents
• Rt Click on “Plane Geometry”
• Choose “Circle”
• Radius 8 mm, Center on (0,0)
• Build all
Choose Circle as Reference Edge
• Rt Click on “Multi-Turn Coil Domain”
• Choose “Edges” > Reference Edge”
• Pick the Broom Icon to Clear the list
• Pick the mid-line circular edges just
drawn on the top of the coil domain
Mesh
• Highlight “Mesh”
• Change “Size” to “Fine”
• Build All
Solve Magnetic Field
• Rt Click on Study 1
• Hit “Compute”
• Rt click on “Model 1”
• Choose “AC/DC” >
“Charged Particle Tracing”
•
•
•
•
Choose the blue “Next” arrow
Choose “Time Dependant”
Deselect “Solve For” Magnetic Fields
Pick Finish Flag
Note: You need an additional study since particle tracing is transient
whereas the magnetic analysis was stationary.
Particles only in Air Domain
• Highlight “Charged Particle Tracing”
• Go to “Domain” section
• Pick “Clear Selection” button (the broom)
• Choose only Air domain, (domain #1)
Add Lorenz Magnetic Force to Particles
• Rt Click on “Charged Particle Tracing”
• Choose “Magnetic Force”
• Select only the Air Domain (#1)
• Under Magnetic Force selection for B list
• Choose “Magnetic flux density mf/mf”
Note Particle Properties
• Under “Charged Particle Tracing”
• Highlight “Particle Properties 1”
Leave as Default values:
• Particle mass: me_const
(mass of an electron)
• Charge Number: -1
BC: Particle Inlet
• Rt Click on “Charged Particle Tracing”
• Choose “Inlet”
• Select Indicated Boundary (#40)
• Choose “Projected Plane Grid
• Change “Initial position” to “Density”
• Set “N” to “500”
Set Initial Velocity:
• 0
:x
• 0
:y
• 1.33e7[m/s] : z
Walls – Leave as “Freeze”
• Under “Charged Particle Tracing”
• Highlight “Wall 1”
• Leave “Wall Condition” as default
“Freeze”
That way we can get statistics later
Assign Stationary Solver to Magnetics only
• Expand “Study 1”
• Highlight “Step 1: Stationary”
• In the “Physics Selection”:
• Deselect “Charged Particle Tracing”
Assign Transient Solver to Particle Tracing
•
•
•
•
Expand “Study 2”
Highlight “Step 1: Time Dependant”
In the “Physics Selection”:
Make sure “Magnetic Fields” is deactivated
• Expand the “Values of Dependent
Variables” section
• Select “Values of variables not solve for”
• Method: “Solution”
• Study: “Study 1, Stationary”
• Stationary: “Automatic”
Note: This uses the magnetic solution obtained in study 1
Set Times and Solve
• Highlight “Step 1: Time Dependant”
•
•
•
•
•
Choose the “Range” button
Set to “Number of Values”
Start: “0”
Stop: “5e-9”
Number of Values: “25”
• Rt Click on Study 2 > Hit Compute
• Expand Particle Trajectories (cpt)
• Highlight “Particle Trajectories 1”
• Change “Type” to “Line”
• Plot
• Rt Click on “Particle
Tracing”
• Choose “Slice Plot”
Change to
• Plane Type: Quick
• Plane: zx-plane
• Planes: 1
• Leave default expression:
mf.normB (magnetic flux
density)
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```
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