Numerical Analyses for ElectronagneticsNumerical Analyses for Electronagnetics Takuichi Hirano (RA)...
Transcript of Numerical Analyses for ElectronagneticsNumerical Analyses for Electronagnetics Takuichi Hirano (RA)...
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June 5, 2002 Tokyo Institute of Technology
No. 1MoM
Numerical Analysesfor Electronagnetics
Takuichi Hirano (RA)Ando & Hirokawa lab.
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June 5, 2002 Tokyo Institute of Technology
No. 2MoM
Differential Equations
iEH
HE
=+×∇
=+×∇
ωε
ωµ
j
jand
0Maxwell’s Equation:
0or
0
22
22
=+∇
=+∇
HH
EE
k
k
qk
k
−=+∇
−=+∇
φφ
µ
22
22
andiAA
V
S
i
Helmholtz’s Equation:
HE,straightforward
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June 5, 2002 Tokyo Institute of Technology
No. 3MoM
Boundary Condition
V
S
i
Boundary condition
微分方程式+境界条件で解が一意に決定!
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June 5, 2002 Tokyo Institute of Technology
No. 4MoM
D.E. + B.C.
Example
xxfdxd
=)(22
21
3
6)( CxCxxf ++=
1)1(0)0(
)(22
==
=
ff
xxfdxd
D.E. (Differential Equation)
D.E. +B.C. (Boundary Condition)
0)0( 2 == Cf
65,1
61)1( 11 ==+= CCf
( )56
)( 2 += xxxf
2階編微分方程式→ 境界値問題(放物型、楕円型、双曲型)
x
f(x)
O 1
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June 5, 2002 Tokyo Institute of Technology
No. 5MoM
Maxwell’s Eq. + B.C.
VS
Boundary condition
マクスウェルの方程式+境界条件で解が一意に決定
iEH
HE
=+×∇
=+×∇
ωε
ωµ
j
jand
0
ある境界条件の下で微分方程式を解きたい!
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June 5, 2002 Tokyo Institute of Technology
No. 6MoM
Numerical Methods
① モーメント法 (Moment Method, Method of Moments, MoM)
・境界のみに未知数を配置
② 有限要素法 (Finite Element Method, FEM)
・空間を細かく分割し、空間全体に未知数を配置・微分方程式を直接解くのでなく、汎関数の極値問題 に置き換えて解く。
③ 時間領域差分法 (FDTD 法, Finite Difference Time Domain method)
・マクスウェルの方程式を差分化して電磁界の伝播 をシミュレートする
Variational method,EMF(Electro Motive Force) method,ICT(Improved Circuit Theory)etc.
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June 5, 2002 Tokyo Institute of Technology
No. 7MoM
モーメント法Method of Moments (MoM)
Moment Method
zh− h0
a
0z 1z 2z 3z 1−Nz Nz 1+Nz
1j 2j 3j Njz∆
Source
Observation
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June 5, 2002 Tokyo Institute of Technology
No. 8MoM
History
1968R. F. Harrington, “Field Computation by Moment Methods”,IEEE Press, New York, 1993
E. Hallen,K. K. Mei,C. T. Tai
1967 (Harrington)R. F. Harrington, “Matrix Methods for Field Problems”,Proc. IEEE, vol. 55, pp.136-149, February 1967
R. F. Harrington, and J. R. Mautz, “Straight Wires With Arbitrary Excitation and Loading”,IEEE Trans. Antenna and Propagat., vol. AP-15, no. 4, pp. 502-514, July 1965
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June 5, 2002 Tokyo Institute of Technology
No. 9MoM
Method of Moments
積分方程式の導出
未知分布量の離散化
ポイントマッチング
行列方程式を解く
モーメント法(ガラーキン法等)
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June 5, 2002 Tokyo Institute of Technology
No. 10MoM
Example: Charged Wire
a− ax
y
z
V
∞
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June 5, 2002 Tokyo Institute of Technology
No. 11MoM
Integral Equation
∫ −= baron 4)()( s
so
so drrr
rrπερφ
a− ax
)( xρ
or
Vd sso
s =−∫ baron 4)( rrr
rπερ
or は導体棒上
Unknown charge distribution
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June 5, 2002 Tokyo Institute of Technology
No. 12MoM
Expansion
0xa=− Nxa =x
)( xρ
1x 2x 3x 1−nx nx 1−Nx
1 2 3 n N
∑=
=N
nnn xfax
1)()(ρ
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June 5, 2002 Tokyo Institute of Technology
No. 13MoM
Point Matching
Vdxfa sso
N
nnn =−∫ ∑=baron 1 4
1)( rrrπε
Vdxfa sso
snN
nn =−∫∑= baron1 4
)( rrrπε
Vdxxx
xfa nns
x
xx sso
snN
nn =−∫∑ −== 1 4
)(1 πε 0xa=− Nxa =
x
)( xρ
1x 2x 3x 1−nx nx 1−Nx
1 2 3 n N
0xa=− Nxa =x
)( xρ
1x 2x 3x 1−nx nx 1−Nx
1 2 3 n N
observation point
21 nnn
oxxx += −
Vdxxx
a
Vdxxx
a
n
ns
n
ns
x
xx ss
No
N
nn
x
xx sso
N
nn
=−
=−
∫∑
∫∑
−
−
==
==
1
1
41
41
1
11
πε
πεM
=
V
V
a
a
zz
zz
NNNN
N
MM
L
MOM
L 1
1
111
∫−= −
= nns
x
xx ss
mo
mn dxxx
z1 4
1πε
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June 5, 2002 Tokyo Institute of Technology
No. 14MoM
Example
-0.4 -0.2 0 0.2 0.4Position HmL2468
101214
eniLegrahc
ytisnedHCpêmL
maVV 1,1 ==m001.0
V
∞
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June 5, 2002 Tokyo Institute of Technology
No. 15MoM
Method of Moments (MoM)
source observation
sr or
source observation
sr or
(a) 重み付けしない場合
(b) 重み付けする場合
ここでは境界条件が満足される保証はない
重み関数
0xa=− Nxa =x
)( xρ
1x 2x 3x 1−nx nx 1−Nx
1 2 3 n N
0xa=− Nxa =x
)( xρ
1x 2x 3x 1−nx nx 1−Nx
1 2 3 n N
observation point
Vd sso
s =−∫ baron 4)( rrr
rπερ
Vdxdxxx
xfxwa
Vdxdxxx
xfxwa
N
No
n
ns
o
n
ns
x
xx
x
xx osso
snoN
N
nn
x
xx
x
xx osso
sno
N
nn
=−
=−
∫ ∫∑
∫ ∫∑
− −
−
= ==
= ==
1 1
1
0 1
4)()(
4)()(
1
11
πε
πεM
Vdxxx
a
Vdxxx
a
n
ns
n
ns
x
xx ss
No
N
nn
x
xx sso
N
nn
=−
=−
∫∑
∫∑
−
−
==
==
1
1
41
41
1
11
πε
πεM
● 全体に境界条件を適用する
(重み付け)
観測するときに工夫する
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June 5, 2002 Tokyo Institute of Technology
No. 16MoM
Basis functions
-0.4 -0.2 0.2 0.4x
-1
-0.5
0.5
1
1.5
fHxL-2 -1 1 2 x
-2
-1
12
3
4fHxL
(a) パルス関数
(b) ルーフトップ関数
全域基底関数 (Entire domain basis functions)
テイラー展開 (Taylor’s expansion)
フーリエ級数展開 (Fourier series expansion)
局所基底関数 (local basis functions)
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June 5, 2002 Tokyo Institute of Technology
No. 17MoM
Example: Dipole Antenna
zh− h0a
zh− h0
Electric field
a
d
iE
2/d2/d−
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June 5, 2002 Tokyo Institute of Technology
No. 18MoM
∫ ⋅= wire sssoeeo drrJrrGrE )();()({ } 0)()()(ˆ =+⋅ oioot rErEr
Integral Equation
z
h− h0
a
)(zJ
x
y
z
)()(ˆ)();()(ˆ oi
owire sssoeeotdt rErrrJrrGr ⋅−=⋅⋅ ∫
sr
or
(ro is on the wire)
グリーン関数
t̂
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June 5, 2002 Tokyo Institute of Technology
No. 19MoM
Expansion
x
y
zor
srt̂
zh− h0
a
0z 1z 2z 3z 1−Nz Nz 1+Nz
1j 2j 3j Njz∆
Source
Observation
∑=
=N
nnna
1)()( rJrJ
)()(ˆ)();()(ˆ1
oi
ossnsoee
N
nno tdat
n
rErrrJrrGr ⋅−=⋅⋅ ∫∑ Γ=
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June 5, 2002 Tokyo Institute of Technology
No. 20MoM
MoM (Weighting, Averaging)
x
y
zor
srt̂
zh− h0
a
0z 1z 2z 3z 1−nz nz 1+nz
njmj
22 )( soso zza −+=−rr
z∆
Source Observation
sr or
{ }
{ }{ }∫
∫ ∫∑
Γ
Γ Γ=
⋅⋅−=
⋅⋅⋅
m
m n
ooi
oomo
ossnsoee
N
nnoomo
dtt
ddatt
rrErrJr
rrrJrrGrrJr
)()(ˆ)()(ˆ
)();()(ˆ)()(ˆ1
∫∫ ∫∑ ΓΓ Γ=
⋅−=
⋅⋅
mm noo
iomossnsoee
N
nnom ddda rrErJrrrJrrGrJ )()()();()(
1
∫∫ ∫∑ ΓΓ Γ=
⋅−=⋅⋅mm n
ooi
omossnsoeeom
N
nn ddda rrErJrrrJrrGrJ )()()();()(
1
{ })(ˆˆ)(ˆ)( omotmom ttJt rJrrJ ⋅==
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June 5, 2002 Tokyo Institute of Technology
No. 21MoM
Matrix Form
x
y
zor
srt̂
zh− h0
a
0z 1z 2z 3z 1−nz nz 1+nz
njmj
22 )( soso zza −+=−rr
z∆
Source Observation
sr or
=
NNNNN
N
V
V
a
a
ZZ
ZZMM
L
MOM
L 11
1
111
neemossnsoeeommnm n
ddZ JGJrrrJrrGrJ∫ ∫Γ Γ =⋅⋅= )();()(
imoo
iomm
mdV EJrrErJ −=⋅−= ∫Γ )()(
観測点を基底関数より多くしたら???条件過剰の方程式を解く問題となる。(ムーア・ペンローズの一般逆作用素、QR分解, SVD?)
)( on rE
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June 5, 2002 Tokyo Institute of Technology
No. 22MoM
Half-wavelength Dipole
-1 -0.75 -0.5 -0.25 0.25 0.5 0.75 1
-1
-0.75
-0.5
-0.25
0.25
0.5
0.75
1
-0.2 -0.1 0 0.1 0.2Position HλL00.002
0.004
0.006
0.008
0.01
edutilpmAHAL
-0.2 -0.1 0 0.1 0.2Position HλL-38-36-34-32-30-28-26
esahPHgedL
Infinitesimal dipole
遠方界指向性
電流分布
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June 5, 2002 Tokyo Institute of Technology
No. 23MoM
Method of Moments
積分方程式の導出
未知分布量の離散化
ポイントマッチング
行列方程式を解く
モーメント法(ガラーキン法等)
)()(),( ygdxxfyxKb
a=∫
Fredholm第一種積分方程式 (参考: 寺沢寛一、「数学概論」、岩波書店)
∑=
=N
nnn xfaxf
1
)()(
∫∫ ∫∑ ΓΓ Γ=
=mm n
dyygywdydxxfyxKywaN
nn )()()(),()(
1
=
NNNNN
N
V
V
a
a
ZZ
ZZMM
L
MOM
L 11
1
111
積分方程式の数値解法!
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June 5, 2002 Tokyo Institute of Technology
No. 24MoM
References
[1] R. F. Harrington, “Field Computation by Moment Methods”, IEEE Press,New York, 1992
[2] C. A. Balanis, “Antenna Theory”, John Wiley & Sons, Inc., 2nd ed., 1997[3] W. L. Stutzman and G. A. Thiele, “Antenna Theory and Design”, John Wiley & Sons, Inc.
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June 5, 2002 Tokyo Institute of Technology
No. 25MoM
Conclusion
Fine