Kein Folientitel

1
Finite Elements in Electromagnetics2. Static
fields
Oszkár Bíró IGTE, TU Graz Kopernikusgasse 24Graz,
Austria email biro_at_igte.tu-graz.ac.at
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Overview

• Maxwells equations for static fields
• Static current field
• Electrostatic field
• Magnetostatic field

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Maxwells equations for static fields
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Static current field (1)
n voltages between the electrodes are given
or
or
n currents through the electrodes are given
i 1, 2, ..., n
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Symmetry
Static current field (2)
GE0 may be a symmetry plane
A part of GJ may be a symmetry plane
6
Interface conditions
Static current field (3)
Tangential E is continuous
Normal J is continuous
7
Network parameters (ngt0)
Static current field (4)
n1
U1 is prescribed and
or
I1 is prescribed and
ngt1
or
i 1, 2, ..., n
i 1, 2, ..., n
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Static current field (5)
Scalar potential V
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Static current field (6)
Boundary value problem for the scalar potential V
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Static current field (7)
Operator for the scalar potential V
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Static current field (8)
Finite element Galerkin equations for V
i 1, 2, ..., n
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High power bus bar
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Finite element discretization
14
Current density represented by arrows
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Magnitude of current density represented by colors
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Static current field (9)
Current vector potential T
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Static current field (10)
Boundary value problem for the vector potential T
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Static current field (11)
Operator for the vector potential T
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Static current field (12)
Finite element Galerkin equations forT
i 1, 2, ..., n
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Current density represented by arrows
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Magnitude of current density represented by colors
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Electrostatic field (1)
n voltages between the electrodes are given
or
n charges on the electrodes are given
i 1, 2, ..., n
on n1 electrodes GE GE0GE1GE2 ... GEi ...
GEn
on the boundary GD
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Electrostatic field (2)
Symmetry
GE0 may be a symmetry plane
A part of GD (s0) may be a symmetry plane
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Electrostatic field (3)
Interface conditions
Tangential E is continuous
Special case s0
Normal D is continuous
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Electrostatic field (4)
Network parameters (ngt0)
n1
U1 is prescribed and
or
Q1 is prescribed and
ngt1
or
i 1, 2, ..., n
i 1, 2, ..., n
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Electrostatic field (5)
Scalar potential V
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Electrostatic field (6)
Boundary value problem for the scalar potential V
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Electrostatic field (7)
Operator for the scalar potential V
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Electrostatic field (8)
Finite element Galerkin equations for V
i 1, 2, ..., n
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380 kV transmisson line
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380 kV transmisson line, E on ground
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380 kV transmisson line, E on ground in presence
of a hill
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Magnetostatic field (1)
n magnetic voltages between magnetic walls are
given
or
or
n fluxes through the magnetic walls are given
i 1, 2, ..., n
on n1 magn. walls GE GE0GE1GE2 ... GEi
... GEn
on the boundary GB
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Magnetostatic field (2)
Symmetry
GH0 (K0) may be a symmetry plane
A part of GB (b0) may be a symmetry plane
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Magnetostatic field (3)
Interface conditions
Special case K0
Tangential H is continuous
Normal B is continuous
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Magnetostatic field (4)
Network parameters (ngt0), J0
n1
Um1 is prescribed and
or
Y1 is prescribed and
ngt1
or
i 1, 2, ..., n
i 1, 2, ..., n
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Magnetostatic field (5)
Network parameter (n0), b0, K0, J?0
Inductance
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Magnetostatic field (6)
Scalar potential F, differential equation
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Magnetostatic field (7)
Scalar potential F, boundary conditions
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Magnetostatic field (8)
Boundary value problem for the scalar potential F
Full analogy with the electrostatic field
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Magnetostatic field (9)
Finite element Galerkin equations for F
i 1, 2, ..., n
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Magnetostatic field (10)
In order to avoid cancellation errors in computing
T0 should be represented by means of edge
elements
since
and hence T0 and gradF (n) are in the same
function space
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Magnetostatic field (11)
Magnetic vector potential A
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Magnetostatic field (12)
Boundary value problem for the vector potential A
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Magnetostatic current field (13)
Operator for the vector potential A
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Magnetostatic field (14)
Finite element Galerkin equations for A
i 1, 2, ..., n
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Magnetostatic field (15)
Consistence of the right hand side of the
Galerkin equations