Integrating the Gradient of the Thin Wire Kernel

2008·0.17 MB·English

by  Champagne, Nathan J.

Integrating the Gradient of the Thin Wire Kernel

Nathan J. Champagne(1) and Donald R. Wilton(2)
(1) ESCG, Houston, TX 77258–8477, USA
(2) University of Houston, Houston, TX 77204–4005, USA

Introduction

When the source and observation points are close, the potential integrals over
wire segments involving the wire kernel are split into parts to handle the singular
behavior of the integrand [1]. The singularity characteristics of the gradient of the
wire kernel are different than those of the wire kernel, and the axial and radial
components have different singularities. The characteristics of the gradient of the
wire kernel are discussed in [2].

Integration of the Gradient of the Wire Kernel

To evaluate the near electric and magnetic fields of a wire, the integration of the
gradient of the wire kernel needs to be calculated over the source wire. Since the
vector bases for current have constant direction on linear wire segments, these
integrals reduce to integrals of the form
z
∫ U B(ξ,ξ)∇K(ρ,z−z′)dz′
1 2
z
L
=⎡∫z−Δz1+∫z +∫z+Δz2+∫zU ⎤B(ξ,ξ)∇K(ρ,z−z′)dz′ (1)
⎢⎣ z z−Δz z z+Δz ⎥⎦ 1 2
L 1 2
=I +I +I +I ,
1 2 3 4
whereB(ξ,ξ) is a basis function in normalized segment variables ξ,ξ =1−ξ
1 2 1 2 1
for scalar current, ∇K(ρ,z−z′) is the gradient of the wire kernel, z is the
projection of the observation point onto the wire, ρ is the perpendicular distance
from the wire axis to the observation point, and Δz and Δz are defined as in [1].
1 2

On the wire surface, the gradient of the wire kernel ∇K(ρ,z−z′) is dominated by
a near-delta function behavior in a neighborhood (Δz and Δz ) about the
1 2
projection of the observation point onto the segment (ξ0,ξ0) [2]. The integrals I
1 2 2
and I are designed to account for this behavior. Outside this neighborhood, the
3
gradient of the wire kernel has a 1 R3 behavior that is treated in I and I .
1 4

The first integral,
z−Δz
I = ∫ 1B(ξ,ξ)∇K(ρ,z−z′)dz′, (2)
1 1 2
z
L
has an integrand dominated by a 1 R3 behavior. This behavior is canceled by
letting
3
dz′= R3 du =⎡(z−z′)2 +(ρ+a)2⎤2 du, (3)
max ⎣ ⎦
or, upon integrating,
z′−z ΔL(ξ −ξ0)
u = = 2 2 and z′= z+(ρ+a)2 R u. (4)
(ρ+a)2 R (ρ+a)2 R max
max max
The use of (3) and (4) allows (2) to be rewritten as
I = ∫u1U B(ξ,ξ)∇K(ρ,z−z(k))R3 du
1 1 2 max
u
1L
≈ΔL∑K1 ⎡⎢wkGL(u1U −u1L)R1(k)⎤⎥B(ξ(k),ξ(k))∇K(ρ,z−z(k)), (5)
k=1 ⎢⎣ ΔL ⎥⎦ 1 2
where
ΔLξ0 ΔLmax(Δξ,−ξ0)
u =− 2 , u =− 1 1 , (6)
1L (ρ+a)2 R 1U (ρ+a)2 R
max max
3
⎛ ρ+a ⎞
R1(k) = Rm3ax =⎜ ⎟ , (7)
2
⎜ 1−⎡u(k)(ρ+a)2⎤ ⎟
⎝ ⎣ 1 ⎦ ⎠
u(k) = u ξGL(k) +u ξGL(k), z(k) = z+(ρ+a)2 R u(k), (8)
1 1L 1 1U 2 max 1
(ρ+a)2 Δz
ξ(k) =ξ0 + R u(k), (ξ(k) =1−ξ(k)), Δξ = 1 =min(ξ0,Δξ) (9)
2 2 ΔL max 1 1 2 1 ΔL 2
and Δξ=a ΔL in this work. The integral I is similarly evaluated.
4

The near-delta function behavior (ρ near but not equal to a) present in the second
integral I is accounted for by letting
2
dz′= R2 du =⎡(z−z′)2 +(ρ−a)2⎤du, (10)
min ⎣ ⎦
which yields, after integrating,
1 ⎛ z′−z⎞ 1 ⎡ΔL(ξ −ξ0)⎤
u = tan−1⎜ ⎟= tan−1⎢ 2 2 ⎥ (11)
ρ−a ⎝ρ−a⎠ ρ−a ⎢ ρ−a ⎥
⎣ ⎦
and
z′= z+(ρ−a)tan⎡(ρ−a)u⎤. (12)
⎣ ⎦
Hence, I may now be written as
2
I =∫u2U B(ξ,ξ)∇K(ρ,z(k))R2 du
2 1 2 min
u
2L
≈ΔL∑K2 ⎡⎢wkGL(u2U −u2L)R2(k)⎤⎥B(ξ(k),ξ(k))∇K(ρ,z−z(k)), (13)
k=1 ⎢⎣ ΔL ⎥⎦ 1 2
where
1 ⎛ΔLΔξ⎞
u =− tan−1 1 , u =0, (14)
⎜ ⎟
2L ρ−a ⎝ ρ−a ⎠ 2U
R(k) = R2 =(ρ−a)2sec2 ⎡(ρ−a)u(k)⎤, (15)
2 min ⎣ 2 ⎦
u(k) = u ξGL(k) +u ξGL(k), z(k) = z+(ρ−a)tan⎡(ρ−a)u(k)⎤, (16)
2 2L 1 2U 2 ⎣ 2 ⎦
and
(ρ−a)
ξ(k) =ξ0 + tan⎡(ρ−a)u(k)⎤, (ξ(k) =1−ξ(k)). (17)
2 2 ΔL ⎣ 2 ⎦ 1 2
The integral I is evaluated using the same approach as I .
3 2

The sample points ξ(k) are given in (9) and (17), while the bracketed quantities in
2
(5) and (13), represent the new weights. These may be used in typical numerical
quadrature routines.

Results

The near field of a 1-meter wire dipole with a radius of 1 mm is calculated and
compared with results using EIGER [3]. The dipole is modeled using 20 linear
segments and lies on the z axis. It is exited at the center with a unit-strength
voltage source at 300 MHz. The magnetic field is sampled 1 mm from the dipole
surface and is shown in Figs. 1 and 2. There is good agreement between EIGER
and the data generated using this formulation.

Summary and Conclusions

A formulation for integrating the gradient of the thin wire kernel is presented.
This approach employs a new expression for the gradient of the thin wire kernel
derived from a recent technique for numerically evaluating the exact thin wire
kernel. This approach should provide essentially arbitrary accuracy and may be
used with higher-order elements and basis functions using the procedure
described in [4].

References

[1] D. R. Wilton and N. J. Champagne, “Evaluation and integration of the thin wire
kernel,” IEEE Trans. Antennas Propagation, vol. 54, no. 4, pp. 1200–1206, 2006.
[2] D. R. Wilton and N. J. Champagne, “Evaluating the gradient of the thin wire
kernel,” 2008 IEEE AP-S International Symposium and USNC/URSI Radio Science
Meeting, submitted.
[3] R. M. Sharpe, J. B. Grant, N. J. Champagne, W. A. Johnson, R. E. Jorgenson,
D. R. Wilton, W. J. Brown, and J. W. Rockway, “EIGER: Electromagnetic
Interactions GEneRalized,” 1997 IEEE AP-S International Symposium and North
American URSI Radio Science Meeting, Montreal, Canada, July 1997, pp. 2366–
2369.
[4] N. J. Champagne, D. R. Wilton and J. D. Rockway, “The analysis of thin wires
using higher-order elements and basis functions,” IEEE Trans. Antennas
Propagation, vol. 54, no. 12, pp. 3815–3821, 2006.

0.14
0.12
0.1
]
m
/ 0.08
A
[

|
y 0.06
H
|
0.04
Hy (EIGER)
0.02
Hy (20,p=0)
0
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Position [m]

Fig. 1. The magnitude of the y component of the magnetic field 1 mm from the wire
dipole.

180
120
]
s
e
e
r 60
g
e Hy (EIGER)
D
[ 0 Hy (20,p=0)

e
s
a
h -60
P

y
H -120
-180
-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5
Position [m]

Fig. 2. The phase of the y component of the magnetic field 1 mm from the wire dipole.