1
EE256 Numerical Electromagnetics
H. O. #6
Marshall
24 June 2008
Summer 2008
LECTURE 5
5.1
FDTD EXPRESSIONS IN THREE DIMENSIONS
The FDTD expressions for Maxwell’s Equations in three dimensions are given in Section 3.6.3 of the text
book,
Computational Electrodynamics
, by Taflove and Hagness. However, noting that the placement of the
electric and magnetic fields on the FDTD cell is somewhat different between the Lecture Notes and the text,
we provide expressions for one electric field and one magnetic field component.
y
y
x
z
x
x
y
z
y
z
x
x
z
y
x
y
(
i,j,k
)
(
i
1
,j
+1
,k)
(
i
1
,j
+1
,k
+1)
(
i,j
+1
,k
)
(
i,j,k
+1)
z
z
Figure 5.1:
Placement of electric and magnetic field components in a threedimensional
staggered mesh, known as the Yee cell.
H
z
n
+1
/
2
i
+1
/
2
,j
+1
/
2
,k
=
H
z
n

1
/
2
i
+1
/
2
,j
+1
/
2
,k
[5.1
a
]
+
Δ
t
μ
i
+1
/
2
,j
+1
/
2
, k
E
x
n
i
+1
/
2
,j
+1
,k
E
x
n
i
+1
/
2
,j,k
Δ
y

E
y
n
i
+1
,j
+1
/
2
,k
E
y
n
i,j
+1
/
2
,k
Δ
x
E
z
n
+1
i,j,k
+1
/
2
=
E
z
n
i,j,k
+1
/
2
[5.1
b
]
+
Δ
t
i,j,k
+1
/
2
H
y
n
+1
/
2
i
+1
/
2
,j,k
+1
/
2
 H
y
n
+1
/
2
i

1
/
2
,j,k
+1
/
2
Δ
x

H
x
n
+1
/
2
i,j
+1
/
2
,k
+1
/
2
 H
x
n
+1
/
2
i,j

1
/
2
,k
+1
/
2
Δ
y
Note that the above expressions are for the case where we have no electric or magnetic current sources,
J
i
or
M
i
. Inclusion of the sources is straightforward, as shown in Section 3.6.3 of the textbook.
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5.2
FDTD ALGORITHM FOR LOSSY MEDIA
In the previous lecture, we derived the 1D and 2D FDTD algorithms by neglecting the loss terms in the
two curl equations. Incorporation of these losses into the FDTD algorithm is straightforward, as discussed
below. We start with the 1D case, which we pose in terms of uniform transmission line equations, noting
that the algorithms we derive for
V
and
I
are directly applicable for the 1D Maxwell’s equation cases by
simple substitution of variables.
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 '09
 Electromagnet, Hz, Lossy Media, fdtd algorithm, FDTD algorithms

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