Modelling current flows in three-dimensional conductive and dielectric bodies
View Patent ↗Current flows in a three-dimensional conductive or dielectric body embedded in a substrate of a different material are modelled in terms of surface currents induced in planar surfaces bounding the body and composed of the same material as the substrate, the surfaces having appropriate values for surface impedance.
1. A method of modelling current flows in a body in three-dimensions embedded in a medium composed of a first material, the body being composed of a second material, comprising:
defining a plurality of planar surfaces that together surround and include the body;
deriving a value for surface admittance of the plurality of planar surfaces; and
modelling current flows in the body in terms of surface currents induced on the planar surfaces of the body composed of the first material and having the derived value for surface admittance
wherein a value for surface admittance is derived in accordance with the expression
u
n
×
J
s
(
r
)
=
u
n
×
Y
s
[
E
(
r
)
]
=
u
n
×
∫
∫
S
ⅆ
S
′
Y
_
_
s
(
r
,
r
′
)
·
E
(
r
′
)
where u n is the unit vector normal to the planar surface, E(r) is the electric field at the surface, J s (r) is the induced surface current at the surface, Y s [ ] is the surface admittance operator that relates the electric field with the induced surface current at the surface and Y s with two overbars is the integral kernel in the boundary surface integral representation of the surface admittance operator.
2. The method of claim 1 , wherein the body is composed of an electrically conductive material.
3. The method of claim 1 , wherein the body is composed of a dielectric material.
4. The method of claim 1 , wherein the medium is composed of a stack of different dielectric material layers.
5. The method of claim 1 , further comprising obtaining a discrete representation of the surface admittance operator for a one-dimensional approximation of the body in accordance with the expression
[
E
1
E
2
]
=
[
Z
s
,
s
Z
s
,
m
Z
s
,
m
Z
s
,
s
]
·
[
J
s
,
1
J
s
,
2
]
with
Z
s
,
s
=
1
2
(
Z
1
+
Z
2
)
Z
s
,
m
=
1
2
(
Z
1
-
Z
2
)
and
Z
1
=
Z
c
,
1
Z
c
,
2
Z
c
,
1
tanh
(
jk
c
,
2
t
2
)
-
Z
c
,
2
tanh
(
jk
c
,
1
t
2
)
Z
2
=
Z
c
,
1
Z
c
,
2
Z
c
,
1
coth
(
jk
c
,
2
t
2
)
-
Z
c
,
2
coth
(
jk
c
,
1
t
2
)
where t is the thickness of the one-dimensional approximation of the body, Z c,i is the characteristic impedance and k c,i is the characteristic propagation constant for plane wave propagation in the medium i.
6. The method of claim 1 , further comprising obtaining a discrete representation of the surface admittance operator for a two-dimensional approximation of the body in accordance with the expression
[ E]=[Z s ]·[I]
with [ Z s ]=[[Z 2 ] −1 ·[dZ 2 ]−[Z 1 ] −1 ·[dZ 1 ]] −1
where [E] represents the column vector of the discrete electric field coefficients and [I] represents the column vector of the discrete surface current coefficients.
7. The method of claim 1 , further comprising obtaining a discrete representation of the surface admittance operator for a three-dimensional representation of the body in accordance with the expression
[ E]=[Z s ]·[I]
with [ Z s ]=[[Z 2 ] −1 ·[dZ 2 ]−[Z 1 ] −1 ·[dZ 1 ]] −1 .
8. Apparatus for modelling current flows in a body in three dimensions embedded in a medium composed of a first material, the body being composed of a second material, comprising:
a store arranged to hold data defining a plurality of planar surfaces that together surround and include the body; and
a processor arranged to derive a value for surface admittance of the plurality of planar surfaces, and to model current flows in the body in terms of surface currents induced on the planar surfaces of the body composed of the first material and having the derived value for surface admittance, wherein the processor derives the value for surface admittance in accordance with the expression
u
n
×
J
S
(
r
)
=
u
n
×
Y
S
[
E
(
r
)
]
=
u
n
×
∫
∫
S
ⅆ
S
′
Y
=
S
(
r
,
r
′
)
·
E
(
r
′
)
where u n , is the unit vector normal to the planar surface, E(r) is the electric field at the surface, J s (r) is the induced surface current at the surface, Y s [ ] is the surface admittance operator that relates the electric field with the induced surface current at the surface and Y s with two overbars is the integral kernel in the boundary surface integral representation of the surface admittance operator.