CAD in Magnetics

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CAD in Magnetics

• Ezio Santini
• Università di Roma "La Sapienza"
• Via Eudossiana 18
• 00184 Rome, Italy

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FEM 

• A method for the analysis of continuum media

CONTINUUM MEDIA IN ELECTRIC ENGINEERING
electromagnetic fields
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FEM Objectives

• Numerical evaluation of
• local quantities

(potentials, fields)
Numerical integration
(in Magnetics inductance, losses, linked flux,
back emf, force, torque)
Integral quantities
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MAXWELL EQUATIONS
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Constitutive Relationships
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PMs
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IRON
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FEM Strategy The steps for achieving one
solution where knowledge of fields is needed
1. Field Solution
2. Post-viewing 
3. Post-processing
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1. Field Solution

• Set-up Maxwell equations for the problem
• State the bondary conditions 
• Write out the problem in terms of potentials

Write a system of algebraic equations   Solve the
system of agebraic equations
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2. Post-viewing

• Give a look to the solution
• (equilines for potentials,
• local values for fields)

3. Post-processing
Calculate what you are looking for, starting from
the values of the potentials and/or the
fields  energy inductances flux linkages back
emf torques losses
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The structure of a FEM program
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The capacitance of a waveguide
Maxwell equations
Since the field is irrotational, it can be
expressed as a function of a scalar potential
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The capacitance of a waveguide - 2
and therefore the equation to integrate is
with the adequate boundary conditions for Va and
Vb
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The capacitance of a waveguide - 3
How FEM works

• A geometry is given.
• 2. The region is subdivided into sub-regions (in
  the following, simplexes).This leads to N
  interpolatory nodes and Ne elements.

3. Into each simplex, the potential behaviour is
supposed to be known
where Vi is the potential at a point i and
ai(x,y) is the so-called shape function.
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The capacitance of a waveguide - 4
How FEM works - 2
For a first-order triangle, n3.  ai(x,y) is a
function that has value 1 at node 1, and values 0
at nodes 2 and 3 the so-called shape
function.   It should be noted that, at this
point thereafter, the shape functions (one for
each node of each triangle) are completely known
in an analytical sense, if the coordinates of the
nodes of the triangle are given.
4. To solve the Laplace equation, an alternative
formulation of this latter can be chosen
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The capacitance of a waveguide - 5
How FEM works - 3
Clearly,functional F depends on the N values of
the potential V in the N interpolatory
nodes FF(V1, V2, V3, ...., VN-1, VN)
If the piecewise approximation is substituted in
the latter equation, and if the sommability of
the integrals is exploited
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The capacitance of a waveguide - 6
How FEM works - 4
Again, Ne is the number of simplexes of W, and N
is the number of nodes part of them, however, is
constrained to have a given value, if they lie on
one of the boundaries. Let this number be Nc the
number of unconstrained nodes will therefore be
Nu N - Nc . 
Integral is minimum when the following Nu
equations are simultaneously verified
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The capacitance of a waveguide - 7
How FEM works - 5
After some algebra, the latter equation leads to
a system of linear algebraic equations   S V
G where S is ( Nu, Nu) matrix (the Dirichlet
matrix), V is a (Nu, 1) vector which i-th
component is the potential in the i-th
interpolatory node, and G is a (Nu, 1) vector
(the known terms). V S-1 G
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The capacitance of a waveguide - 8
CAPACITANCE
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FEM IN MAGNETICS
2D PROBLEMS
in the following, Az will be referred at simply
as A.
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Physical Meaning of the vector potential A
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Physical Meaning of the Vector Potential A
A1 A1' A2 A2'
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Physical Meaning of the Vector Potential A
Therefore, the flux between two points (!) is
simply the difference between the vector
potential in the points (per unit
lenght).   Conversely, a line along which the
vector potential is constant is a flux line
These considerations are very useful for the
statement of The Boundary Conditions
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The Boundary Conditions
Fixed values of the potential Neumann
boundaries   Zero values of the normal derivative
of the potential Dirichlet boundaries   Dirichlet
conditions are natural in FEM if nothing is
specified along a boundary, FEM assumes that it
is a Dirichlet boundary.   Periodic behaviour of
the potential binary boundaries.
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The Boundary Conditions
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The linked flux
Two turns parallel-connected
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The linked flux
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Harmonic Fields
             
formally analogous to the Helmholtz
non-homogeneous equation
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FEM in Magneticsa magnetostatic example
1. The initial triangulation (158 nodes)
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A magnetostatic example

• The algorithm is the true link between AutoCAD
  and FEM it is very straightforward.
• Numer of regions is zero.
• Number of regions number of regions 1
• All triangles are scanned.
• The first tringle that do not belongs (yet) to a
  region is in the current region.
• Next triangle belongs to a region NOP.
• Next triangle shares an edge with a triangle of
  the current region the triangle belongs to
  current region.
• All triangles have been scanned.
• Not all triangles have been assigned a region go
  to step 2.

• The subdivision into regions

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The assignement to the regions of material
properties and input quantities
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The potential lines for the vector potential
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• An intermediate solution (1868 nodes)

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The final solution (2924 nodes)
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The evolution of the solution
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The minimum geometry solution
The energy value 1.46175 x 4 5.847
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The comparison between the full geometry and the
minimum geometry solutions
Energy value from minimum geometry 5.847 Nodes
for minimum geometry 2840, that approximately
are equivalent to 1100 nodes in the full geometry
Energy value from full geometry 5.763 Nodes for
full geometry 2924
The difference 1.43
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Inductance evaluation
From magnetic energy
From linked flux
In linear cases, inductance from energy and
inductance from linked flux are equal.
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Back emf evaluation
The initial geometry
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Back emf evaluation
The mesh
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The back emf evaluation
The potential lines of A
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The back emf evaluation
The flux linked with one turn (one pole)
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The back emf evaluation
The flux linked with one turn (two poles)
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The back emf evaluation
The back emf (one turn)
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The back emf evaluation
The Fourier expansion (one turn)
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The back emf evaluation
The back emf (phase)
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The back emf evaluation
The Fourier expansion (phase)
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CONCLUSIONS
definition of the
geometry
nodes and edges
batch process
interactive process
base triangulation
automatic choices
user choices
boundary conditions
input quantities
characteristics of
the materials
The structure of a FEM program
discretization
algebraic solution
error evaluation
new triangulation
the solution
NO
meets the error
mesh refinement
requirements?
YES
post viewing
post processing
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CONCLUSIONS

• FEM programs
• Easy to link to drawing programs
• Fast
• Reliable
• Numerically stable
• Accurate for integral quantities
• Not-accurate for local quantities

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Reference books
P.P. Silvester R.L. Ferrari - Finite
Elements for Electrical Engineers - III Edition,
Cambridge University Press. P. P. Silvester and
D. Lowther - CAD in Magnetics R. Ratnajevaan H.
Hoole - Computer Aided Analysis and Design of
Electromagnetic Devices E.S. Hamdi - Design of
Small Electrical Machines
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CAD in Magneticsend of presentation

• Ezio Santini
• Università di Roma "La Sapienza"
• Via Eudossiana 18
• 00184 Rome, Italy
• ezio_at_elettrica.ing.uniroma1.it
• mobile [phone]