Biomedical Engineering Reference
In-Depth Information
similar to those relating the real and imaginary parts of impedance in general
circuit theory [7].
The structure of Maxwell's equations shows that permittivity and conduc-
tivity are related parameters. To keep it simple, one may say that they express
the link between current density and electric field: When both parameters are
real, the permittivity is the imaginary part while the conductivity is the real
part of this relationship. This can be written as
Jj
=
(
we
+
s
)
E
A m
(1.13)
-
2
When the permittivity is written as complex, there is an ambiguity. There are,
however, too many parameters, as can be seen in the expression
Jj
=
we
(
¢ -
e
¢¢
)
E E
+
s
(1.14)
=¢
j
we
E
++¢¢
(
s
we
)
E
from which it appears that the real part of the relation between the current
and the electric field can be written either as an effective conductivity equal
to
ss
=¢+ ¢
e
S m
(1.15)
-
1
eff
or as an effective imaginary part of permittivity equal to
s
w
s
¢¢
=
F m
(1.16)
-
1
eff
It should be observed that these two expressions are for the conductivity and
permittivity, respectively, and not just the relative ones. Both expressions are
correct and in use. Generally, however, the effective conductivity is used when
characterizing a lossy conductor, while the effective imaginary part of the per-
mittivity is used when characterizing a lossy dielectric. At some frequency, the
two terms are equal, in particular in biological media. As an example, the fre-
quency at which the two terms s¢ and we≤ are equal is in the optical range for
copper, about 1 GHz for sea water, 100 MHz for silicon, and 1 MHz for a humid
soil. Although both expressions are correct, one needs to be careful in such a
case when interpreting the results of an investigation.
One more comment, however, is necessary. It has just been said, and it is
proven by Kramer and Kronig's formulas (1.12), that, if the permittivity varies
as a function of frequency, it must be a complex function. In fact, this is true
of all three electromagnetic parameters: permittivity, conductivity, and per-
meability. Hence, by writing the conductivity as a real parameter in Eqns.
(1.13)-(1.16), we have assumed that it was independent of frequency, which is
about the case of the steady conductivity. If it varies with frequency, then it
has to be written as a complex parameter s¢=s¢+
j
s≤ and Eqns. (1.13)-(1.16)
have to be modified consequently.
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