Time domain surface impedance concept for low frequency ...

d, respectively. The two co-ordinate systems are related as
follows:
x 1 ¼ d 1 a 1 ; x 2 ¼ d 2 a 2 ð13Þ
Equations (9)–(11) with the co-ordinates in (13) are written
in the form
@H xk
X 2
@Z
i¼1
@E xk
@Z
X 2
i¼1
d k
d k
@H Z
Z @x k
@H xi
H xk
d k Z ¼ð 1Þ3 k sE x3 k
; k ¼ 1; 2;
ð 1Þ i d 3 i
¼ sE Z
d 3 i Z @x 3 i
ð14Þ
d k
d k
@E Z
Z @x k
E xk
d k Z ¼ð @H 1Þk x3
m k
1 ; k ¼ 1; 2;
@t
@E xi
ð 1Þ 3 i d 3 i
d 3 i Z @x 3
@H Z
@Z þ X2
d
i¼1 i
d i
@H Z
¼ m 1
i @t
@H xi
X 2
¼ H Z
Z @x i i¼1
4 Dimensionless variables
d i
ð15Þ
ð ZÞ 1 ð16Þ
Following perturbation theory methods, we introduce the
characteristic scale factors for the variables of the problem.
The choice of the scale factors for co-ordinates x 1 ; x 2 ; Z is
evident, namely the characteristic size D ¼ minðd 1 ; d 2 Þ of
the conductor’s surface for x 1 and x 2 , and the penetration
depth d for Z. The quantity t was introduced in (8) as the
ratio 2/o in the case of time-harmonic fields or the incident
pulse duration in the case of transient sources and is
therefore the natural scale factor for time. We denote for
now the scale factors for the electric and magnetic fields as
E and H , respectively.
Let us now switch to the following non-dimensional
variables:
~x k ¼ x k =D; ~t ¼ t=t; ~Z ¼ Z=d; ~E ¼ E=E ; ~H
¼ H=H
ð17Þ
Here and below, the sign ‘B’ denotes non-dimensional
quantities.
Substitution of (17) into (14)–(16) gives
H
d
@ ~H xk
@~Z
d
D
d k
d k
@ ~H Z
d~Z @ ~ x k
¼ ð 1Þ 3 k E s~E x3 k
; k ¼ 1; 2
d k
~H xk
d~Z
ð18aÞ
H X 2
ð 1Þ i d 3 i @ ~H xi
D d
i¼1 3 i d~Z @ ~ ¼ sE ~E Z ð18bÞ
x 3 i
E
D @~E xk d k @~E Z
~E xk
D d @~Z d k d~Z @ ~ x k
d k d~Z
ð19aÞ
¼ ð 1
Þ k H @ ~H x3
m 1
t @~t
k
; k ¼ 1; 2
E X 2
ð 1Þ 3 i d 3 i @~E xi H @ ~H Z
D
d
i¼1
3 i d~Z @ ~ ¼ m 1
x 3 i
t @~t
D @ ~H Z
d @~Z þ X2
i¼1
d i
d i
@ ~H xi
X
d~Z @ ~ ¼ D ~H 2
Z
x i i¼1
d i
ð19bÞ
ð d~Z Þ 1 ð20Þ
Note that not all scale factors in (18)–(20) are actually
independent. Indeed, from (8) it follows that
p
d ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
t=ðsm 1 Þ ¼ t=ðsm 1 D 2 ÞD ¼ pD;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð21Þ
p ¼ d=D ¼ t=ðsm 1 D 2 Þ 1
where the parameter p is proportional to the ratio of the
penetration depth and characteristic size of the conductor’s
surface. The relation between E and H follows from (19):
E ¼ m 1D
t H ð22Þ
Therefore, the total number of basic scale factors is 3, for
example: t, D and H . The practical selection of the scale
factors should be based on the input data of a given
problem [21]. Sometimes the total current is used as input
data. In such cases it is natural to use the characteristic
current I as one of the basic scale factors instead of E or
H . The relation between I and H can be obtained using
the Biot-Savart law
Z
~H ¼ ð4pÞ 1 ~I ~R=R 3 dl ð23Þ
L
Transfer to the non-dimensional variables in (23) gives:
Z
~~HH ¼ I ð4pDÞ 1 ~~I ~
~R=~R 3 d ~ l ð24Þ
L
Thus
H ¼ I = ð4pD Þ ð25Þ
Substitution of (21)–(22) into (18)–(19) gives
p @ ~H xk
@~Z
p 2 X2
i¼1
@~E xk
@~Z
p 2 X2
i¼1
p 2 ~H xk
~d k p~Z
p 2 dk ~ @ ~H Z
~d k p~Z @ ~ ¼ð 1Þ 3 k ~E x3 k
; k ¼ 1; 2;
x k
~
ð 1Þ i d3 i @ ~H xi
~d 3 i ~Z @ ~ ¼ ~E Z
x 3 i
ð26Þ
p~E xk
~d k p~Z
p d ~ k @~E Z
~d k p~Z @ ~ ¼ð 1Þ k p @ ~H x3
x k
@~t
~
ð 1Þ 3 i d3 i @~E xi
~d 3 i ~Z @ ~ ¼ @ ~H Z
x 3 i
@t
@ ~H Z
@~Z
p ~H Z
X 2
i¼1
1
~d i p~Z ¼ p X2
i¼1
~d i @ ~H xi
~d i p~Z @ ~ x i
k
; k ¼ 1; 2;
ð27Þ
ð28Þ
where ~ d k ¼ d k =D; k ¼ 1; 2. Equations (26)–(28) do not
contain the scale factors for the electric and magnetic fields
anymore. The remaining scale factors for the co-ordinates
are included only in the form of the ratio d/D which is the
small parameter of the problem.
5 Expansions in the small parameter
We now apply Laplace’s transform following the rule
ef ðesÞ ¼ R 1
0
ef ðetÞ expð esetÞdet. Here f denotes an arbitrary
function. In the Laplace-domain, (26)–(28) are written
178 IEE Proc.-Sci. Meas. Technol., Vol. 152, No. 4, July 2005