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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,

Ar(r, t) the magnetic vector potential on the ramp of the current rise

(0≤t≤t0)

Ac(r, t) the magnetic vector potential on the constant part of the

current rise (t0≤t)

A0(x, y) the initial condition for A at time t=0,

a(k, n) the magnetic vector potential at node k and at time tn,

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,

an a column vector which holds the nodal values of the magnetic

vector potential at time tn

a = (a1, a2, . . . , aj, . . . , aN)t

B 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,

Dr the radial grid spacing,

Dt increment in time,

Di an open triangle in ,

Di* the closure of Di contained in ,

De the area of the triangle e,

dR(r, 0) a variation of the Dirac delta function,

the boundary of ,

∂R the boundary of the domain R, ∂R = ∂RD ∂RN

∂RN the part of the boundary ∂R where the Neumann boundary condition holds,

∂RD the part of the boundary ∂R where the Dirichlet boundary condtion holds,

E electric field intensity vector,

Ei error term, i= 1, 2, 3,

Ei = ∫∫ ( s∂a/∂t ) ji dx dy

e permitivity (dielectric constant),

F cross-sectional area of the conductor perpendicular

to the z-axis,

Fk 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,

I0 maximum current magnitude,

I0() modified Bessel function of the first kind and zeroth order,

I1() 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.

ji 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,

Je(x, y, t) the z-component of the eddy-current density,

Js(x, y, t) the z-component of the source current density,

Js(x, y, t) = s C(t)

Jr(r, t) the total current density on the ramp of the current rise

(0≤t≤t0)

Jc(r, t) the total current density on the constant part of the current

rise (t0≤t)

J1(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,

K0() 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,

L2() a space of functions f(x, y), (x, y) , such that

f L2() if and only if || f ||2 = ∫∫ | f(x, y) |2 dx dy < ,

li 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 ∂RN.

n index on time, values are all known for time tn;

unknown values correspond to time tn+1,

ni a parameter in an error estimate, 0≤ ni ≤1, i=1, 2, 3,

n the magnetic reluctivity of the medium,

n0 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 and

whose 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 = Sg accounts for the n02 term after it has been integrated by parts

Se denotes the local contribution from integrating over an element

to the global matrix Sg,

Sij = ∫∫ n0 ( jj/∂x ji/∂x + ∂jj/∂y ji/∂y ) dx dy

s(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,

xn nth positive root of J1,

T(x, y, t) temperature,

t time,

to initial time, in our case to = 0,

tn time at node n, tn = to + ∆t * n,

a a polygon over which the basis function is determined, indexed

by a,

q Crank-Nicolson parameter,

q angle in cylindrical coordinates,

Ui = ∫∫ ( -Js(x, y, t) ) ji dx dy

V relates the position of the source current term with the proper element, V = (1/cn+1) U,

wi(x, y) the weighting function corresponding to node i.

(xi, yi) the coordinate of the ith node of element e, and

|| ||H1() the norm in the Hilbert space over .