Data processing: measuring – calibrating – or testing – Measurement system in a specific environment – Electrical signal parameter measurement system
Reexamination Certificate
2002-05-29
2004-08-03
Hoff, Marc S. (Department: 2857)
Data processing: measuring, calibrating, or testing
Measurement system in a specific environment
Electrical signal parameter measurement system
C702S183000, C702S188000
Reexamination Certificate
active
06772076
ABSTRACT:
BACKGROUND OF THE INVENTION
1. Field of the Invention
The present invention relates to the technology of performing an electromagnetic field analysis based on the FDTD (Finite Difference Time Domain) method by dividing an analyzed space into hexahedral cells.
2. Description of the Related Art
The electromagnetic field analysis is performed in various fields on the antenna problem, the scattering of electromagnetic waves, etc. As analyzing methods, the FEM (Finite Element Method), the moment method, the BEM (Boundary element method), etc. have been used. Recently, the FDTD (Finite Difference Time Domain) method, which is an FDM (Finite Difference Method) suggested by K. S. Yee, has attracted much attention because of a number of advantages such as its simplicity in algorithm, high precision, wide application, etc.
In the FDTD method, an electromagnetic field is differentiated in a time domain and an analyzed space, and the behavior of an electromagnetic field is obtained by a numerical analysis based on the geometric arrangement and shape, and the electric (physical) characteristics such as the conductivity, magnetic permeability, etc. of an object to be analyzed. In a time domain, a discrete transformation is performed in a very short time step (&Dgr;t). An analyzed space is divided into cells for the discrete transformation.
FIG. 1
is an explanatory view of the cell dividing an analyzed space. As shown in
FIG. 1
, the cell is set as hexahedrons (normally cubes) each having the lengths of &Dgr;x in the x-axis direction, &Dgr;y in the y-axis direction, and &Dgr;z in the z-axis direction. The magnetic field intensity is placed in the center of each plane, and the field intensity is placed in the center of each side. Thus, the field rotation produces a magnetic field, and the magnetic field rotation produces an electric field by differentiating the magnetic field intensity from the field intensity by half-cell spatially shifting them from each other, thereby directly solving the Maxwell's equation. The cell size is set to be less than {fraction (1/10)} of the shortest wavelength normally defined as a problem.
The electric field and the magnetic field differentiated by spatially shifted from each other are alternately arranged in a time domain as shown in FIG.
2
. That is, the leap-frog algorithm for alternately computing the field intensity and the magnetic field intensity is used. Thus, the magnetic field intensity is obtained from the field intensity, and the field intensity is obtained from the magnetic field intensity, and these processes are alternately performed. The superscripts such as ‘n−½’, ‘n’, etc. shown in
FIG. 2
represent the positions arranged in the time domain. The difference of the time domain is obtained such that the Courant stability condition can be satisfied.
In the FDTD method, the central difference method is used in differentiating the time domain and the analyzed space using the above mentioned leap-frog algorithm. Depending on the field in which an analysis is performed, the arrangements of the electric field and the magnetic field in a time axis or spatial coordinates can be inverse to each other.
The FDTD method is based on the Maxwell's equation, which is also based on the Ampere's circuital law, and the Faraday's law of induction. Each law is represented in the differentiation form and the integral form, and the above mentioned Maxwell's equation refers to a total of the four equations obtained by combining with the Gauss' law on electric field and magnetic field. The Maxwell's equation in the differentiation form is represented as follows using the field intensity E [V/m], the magnetic field intensity H [A/m], the electric flux density D [C/m
2
], the magnetic flux density B [T], the electric charge density &rgr; [C/m
3
], and the current density J [A/m
2
].
rot
H
(
r
,
t
)
=
∂
D
(
r
,
t
)
∂
t
+
J
(
r
,
t
)
(
1
)
rot
E
(
r
,
t
)
=
∂
B
(
r
,
t
)
∂
t
(
2
)
rot
B
(
r
,
t
)
=
0
(
3
)
rot
D
(
r
,
t
)
=
ρ
(
r
,
t
)
(
4
)
where the equation (1) (Ampere's law), the equation (2) (Faraday's law of induction), the equation (3) (Gauss' law in the magnetic field), and the equation (4) (Gauss' law in the electric field) are not independent equations. That is, the equations (3) and (4) are at most used as the evaluation references of a numerical error in the FDTD method, and the equations (1) and (2) are used in formulation. If the formulation is performed by applying the central difference method to the time domain and the spatial coordinate using the algorithm of Yee in the equations (1) and (2), then the following FDTD basic equation can be obtained in a 3-dimensional space.
E
x
n
(
i
+
1
2
,
j
,
k
)
=
K
1
x
(
i
+
1
2
,
j
,
k
)
E
x
n
-
1
(
i
+
1
2
,
j
,
k
)
+
K
2
x
(
i
+
1
2
,
j
,
k
)
×
{
[
H
z
n
-
1
2
(
i
+
1
2
,
j
+
1
2
,
k
)
-
H
z
n
-
1
2
(
i
+
1
2
,
j
-
1
2
,
k
)
]
Δ
z
-
[
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
y
}
(
5
)
H
x
n
+
1
2
(
i
,
j
+
1
2
,
k
+
1
2
)
=
H
x
n
-
1
2
(
i
,
j
+
1
2
,
k
+
1
2
)
-
K
3
x
(
i
,
j
+
1
2
,
k
+
1
2
)
{
[
E
z
n
(
i
,
j
+
1
,
k
+
1
2
)
-
E
z
n
(
i
,
j
,
k
+
1
2
)
]
Δ
z
-
[
E
y
n
(
i
,
j
+
1
2
,
k
+
1
)
-
E
y
n
(
i
,
j
+
1
2
,
k
)
]
Δ
y
}
(
6
)
K
1
x
(
i
+
1
2
,
j
,
k
)
=
1
-
σ
(
i
+
1
2
,
j
,
k
)
Δ
t
2
ϵ
(
i
+
1
2
,
j
,
k
)
1
+
σ
(
i
+
1
2
,
j
,
k
)
Δ
t
2
ϵ
(
i
+
1
2
,
j
,
k
)
(
7
)
K
2
x
(
i
+
1
2
,
j
,
k
)
=
Δ
t
ϵ
(
i
+
1
2
,
j
,
k
)
1
+
σ
(
i
+
1
2
,
j
,
k
)
Δ
t
2
ϵ
(
i
+
1
2
,
j
,
k
)
1
Δ
y
Δ
z
(
8
)
K
3
x
(
i
,
j
+
1
2
,
k
+
1
2
)
=
Δ
t
μ
(
i
,
j
+
1
2
,
k
+
1
2
)
1
Δ
y
Δ
z
(
9
)
where &mgr;, &egr;, and &sgr; respectively indicate the magnetic permeability, permittivity, and electric resistivity.
The characters ‘x’ and ‘y’ added as the subscripts to the respective symbols E and H in the equations (5) and (6) indicate the direction of the field intensity and the magnetic field intensity. In these equations, only the field intensity and the magnetic field intensity assigned in the x direction are shown, but they can also be derived in the y and z directions. The documents of the conventional technology to be referred to in understanding the present invention are: ‘FDTD (Finite Difference Time Domain) method’ by Hano, 2002 National Convention Record IEE Japan, vol. 5, pp 411-414, 2002 (hereinafter referred to as the conventional technology document 1); ‘Development of a General Surge Analysis Program Based on the FDTD Method’ by Noda and Yokoyama, The Transactions of the Institute of Electrical Engineers of Japan, vol. 121-B, No. 5, pp 625-632, 2001 (hereinafter referred to as the conventional technology document 2); ‘A Locally Conformed Finite-Difference Time-Domain Algorithm of Modeling Arbitrary Shape Planar Metal Strips’ by J. Fang and J. Ren; IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 41, NO. 5, MAY 1993 (hereinaft
Iki Hiroyuki
Yamamoto Kazuo
FFC Limited
Greer Burns & Crain Ltd.
Hoff Marc S.
LandOfFree
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