List of Symbols
Symbol Usage
A the magnetic vector potential,
A(r, t) the z-component of the magnetic vector potential at radius r at
time t,
A(x, y, t) the z-component of the magnetic vector potential,
A
r(r, t) the magnetic vector potential on the ramp of the current rise(0≤t≤t
0)A
c(r, t) the magnetic vector potential on the constant part of thecurrent rise (t
0≤t)A
0(x, y) the initial condition for A at time t=0,a(k, n) the magnetic vector potential at node k and at time t
n,a(r, t) the approximation to the magentic vector potential,
a(x, y, t) the approximation to the magnetic vector potential at postion
(x, y) in the Cartesian coordinate system at time t,
a
n a column vector which holds the nodal values of the magneticvector potential at time t
na
= (a1, a2, . . . , aj, . . . , aN)tB
the magnetic induction (flux density) vector,bi = ym - yj
ci = xm - xj in cyclic modulo 3, with the permutation (i, j, m),
c speed of an electromagnetic wave through the medium,
C(tn) = cn an unknown scalar constant which represents the magnitude of
source current divided by s at time tn, for which to solve, ,
and must satisfy the current constraint,
D electric displacement (electric flux density vector),
D relates the conductivities to correspond with the proper terms for the magnetic vector potential, E = D (an+1 - an) / Dt,
—
del,— ¥
curl,—
. div,—
2 the Laplace's operator,D
r the radial grid spacing,D
t increment in time,D
i an open triangle in ¬,D
i* the closure of Di contained in ¬,D
e the area of the triangle e,d
R(r, 0) a variation of the Dirac delta function,∂
¬ the boundary of ¬,∂R the boundary of the domain R, ∂R = ∂R
D » ∂RN∂R
N the part of the boundary ∂R where the Neumann boundary condition holds,∂R
D the part of the boundary ∂R where the Dirichlet boundary condtion holds,E electric field intensity vector,
E
i error term, i= 1, 2, 3,E
i = ∫∫¬ ( s∂a/∂t ) ji dx dye
permitivity (dielectric constant),F cross-sectional area of the conductor perpendicular
to the z-axis,
F
k area per unit radial distance of the area element, Fk=2prk,H
magnetic field intensity vector,h(t) Heaviside function,
I
(t) the given total current in the conductor,I(t) is a piecewise linear continuous function,
I
the imposed current,I
0 maximum current magnitude,I
0() modified Bessel function of the first kind and zeroth order,I
1() modified Bessel function of the first kind and first order,i, j, and m are indices on the vertices of the triangle e arranged in
clockwise order.
j
i the basis function at node i,J the current density vector,
J(x, y, t) the z-component of the total (eddy and source) current density,
J
e(x, y, t) the z-component of the eddy-current density,J
s(x, y, t) the z-component of the source current density,J
s(x, y, t) = s C(t)J
r(r, t) the total current density on the ramp of the current rise(0≤t≤t
0)J
c(r, t) the total current density on the constant part of the currentrise (t
0≤t)J
1(x) modified Bessel function of the first kind and first order,K diffusion constant,
K =
» i=1,...N Di* is a closed convex set whose boundary ∂K is a polygon whose vertices belong to ∂¬K(t) ramp waveform,
K
0() modified Bessel function of the second kind and zeroth order,k nodal point at the element;
indexing begins with 0 and ends with M
L length of the conductor,
L{} Laplace transform,
L
2(¬) a space of functions f(x, y), (x, y)Œ ¬, such thatf
Œ L2(¬) if and only if || f ||2 = ∫∫¬ | f(x, y) |2 dx dy < •,l
i a parameter in an error estimate, 0≤ li ≤1, i=1, 2, 3,M maximal index on the region accounted for by the
conductor,
m
permeability,n the normal to the boundary curve ∂R
N.n index on time, values are all known for time t
n;unknown values correspond to time t
n+1,n
i a parameter in an error estimate, 0≤ ni ≤1, i=1, 2, 3,n
the magnetic reluctivity of the medium,n
0 the magnetic reluctivity of free space,f
electric scalar potential function,F
flux density,p
3.1415...,p Laplace transform parameter of time,
√
n(x, y) a set of basis functions of degree n over ¡a,Y
(x, y) a prescribed function on ∂RD,q a constant which non-dimentionalizes the diffustion equation,
R the real two dimensional space,
¬
a bounded convex set which approximates R andwhose boundary is a piecewise differentiable curve
r radius,
r
volume charge density,r(x, y, t) the residual at postion (x, y) in the Cartesian coordinate
system at time t,
S = S
g accounts for the n0—2 term after it has been integrated by partsS
e denotes the local contribution from integrating over an elementto the global matrix S
g,S
ij = ∫∫¬ n0 ( ∂jj/∂x ∂ji/∂x + ∂jj/∂y ∂ji/∂y ) dx dys
(x, y, T(x, y, t)) the electrical conductivity, varies with the temperature T, s(k, n) the conductivity at node k and at time tn,x
n nth positive root of J1,T(x, y, t) temperature,
t time,
t
o initial time, in our case to = 0,t
n time at node n, tn = to + ∆t * n,¡
a a polygon over which the basis function is determined, indexedby
a,q
Crank-Nicolson parameter,q
angle in cylindrical coordinates,U
i = ∫∫¬ ( -Js(x, y, t) ) ji dx dyV relates the position of the source current term with the proper element, V = (1/c
n+1) U,w
i(x, y) the weighting function corresponding to node i.(x
i, yi) the coordinate of the ith node of element e, and|| ||H
1(¬) the norm in the Hilbert space over ¬.