The analytical expression for the inductance
is obtained from the magnetic
energy [102]
It must be distinguished between three regions: the inner conductor
,
the dielectric between the conductors
, and the outer conductor
.
For
the magnetic flux
and the integral
are given by
For
The current
flows along the inner conductor. The same current returns along
the outer conductor flowing in the opposite direction. Similarly to
(6.4), for
where
,
only the current through the circle inside the integration loop must
be considered
Now the inductance can be obtained from the integral over the entire domain
The resistance
of the conductors is given by
Equations (6.8) and (6.9) are obtained assuming a constant current density
distribution in the conductors. This is true only for low frequencies. The distinction between low and high
frequency is in terms of the skin effect. Thus, whether an operating
frequency is considered as low or high depends also on the dimensions of the
geometries, not only on the frequency itself. At high frequencies for which
skin effect is not negligible
and
are modified to read [102]
where
and
are taken from (6.9),
is the thickness of the outer
conductor (
), and the skin depth
is given by the expression
Notice that (6.10) and (6.11) are valid only, if
is reasonably
small compared to
(
).