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5.2 Theoretical Analysis 171
every H component is surrounded by four circulating E components. The po-
sition of the electric- and magnetic-field vector components is approximately
a cubic unit of the Yee space lattice.
We now rewrite the vector components of (5.1), yieldingthe following
system of six coupled scalar (5.4)–(5.9) [5.15].

1
∂H x ∂E y ∂E z
= − , (5.4)
∂t µ ∂z ∂y
1
∂H y ∂E z ∂E x
= − , (5.5)
∂t µ ∂x ∂z
1
∂H z ∂E x ∂E y
= − , (5.6)
∂t µ ∂y ∂x
∂E x 1 ∂H z ∂H y
= − − σE x , (5.7)
∂t ε ∂y ∂z

∂E y 1 ∂H x ∂H z
= − − σE y , (5.8)
∂t ε ∂z ∂y

∂E z 1 ∂H y ∂H x
= − − σE z . (5.9)
∂t ε ∂x ∂y
Here, from (5.3) we define the function F on (i, j, k) at the time increment n
as
n
F (i, j, k)= F (i∆x, j∆y, k∆z, n∆t) . (5.10)
The space and time derivatives are given as
1
1
n

∂F (i, j, k) F n i + ,j,k − F n i − ,j,k
2
2
= , (5.11)
∂x ∆x
n
∂F (i, j, k) F n+ 1 2 (i, j, k) − F n− 1 2 (i, j, k)
= . (5.12)
∂t ∆t
The space derivative (5.11) has the same form for y, z and the time deriv-
ative (5.12) is given between half an increment and half a decrement. Consid-
ering F as E or H, (5.4)–(5.9) are expressed as the time-steppingexpressions
(5.13)–(5.18).

n+ 1 1 1 n− 1 1 1 ∆t
H x 2 i, j + ,k + = H x 2 i, j + ,k + + 1 1
2 2 2 2 µ i, j + ,k +
2 2
n 1 n 1
E y i, j + ,k +1 − E y i, j + ,k
2
2
×
∆z
E z n i, j, k + 1 2 − E z n i, j +1,k + 1 2
+ ,
∆y
(5.13)