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170 5 Near Field
where ∆x, ∆y, and ∆z are the lattice space increment in the x-, y-, and z-axis,
respectively, and ∆t is the time increment, and i, j, k and n are integrals. Ab-
breviating∆x, ∆y, ∆z, and ∆t, the expression can be rewritten more simply
as
n
F(x, y, z, t)= F (i, j, k), (5.3)
where (i, j, k) are the coordinates of the lattice. The finite-difference expres-
sions for the space derivatives used in the curl operators are central difference
in nature and second-order accurate.
Figure 5.3 shows the space–time chart of the Yee algorithm in one dimen-
sion showingthe use of central differences for space derivatives and leapfrog
for time derivatives. E is positioned at t =(n − 1)∆t, n∆t, (n + 1)∆t,and H
n
is positioned at t =(n − 1/2)∆t, (n +1/2)∆t. In actual case, E is derived
from E n−1 at t =(n−1)∆t and H n−1/2 at t =(n−1/2)∆t and H n+1/2 is de-
n
rived from H n−1/2 and E . Figure 5.4 shows the position of the electric- and
magnetic-field vector components in two dimension. E and H components are
positioned alternately, i.e., H is on every cell edge and E is between cells.
In three-dimensional (3-D) space, E and H components are positioned so
that every E component is surrounded by four circulating H components, and

E n-1 E n E n+1

1 1
n - Dt n + Dt
2 2
t
(n-1)Dt nDt (n+1)Dt

1 1
n- n+
H 2 H 2
Fig. 5.3. Space–time chart of Yee algorithm in one dimension showing the use of
central differences for space derivatives and leapfrog for time derivatives [5.15]

Dy
y
E x
Dx E y
x H z
Fig. 5.4. Positions of electric- and magnetic-field vector components in two dimen-
sion. E and H components are positioned alternately, i.e., H is on every cell edge
and E is between cells [5.15]