shift if the transition time is much shorter than the time constant
and the relaxation is negligible.
The above examples suggest that a short dielectric time constant
(high conductivity) relative to voltage transition times is
preferable; however, high conductivity combined with high field
can cause substantial resistive heating which is proportional to
the conductivity and the square of the field. Such power dissipation
places an upper limit on the acceptable grading material
conductivity. In the case of a bipolar waveform with a 10 ms
transition time, the average power density at the end of each
voltage reversal reaches 42 kW/m3
, and at the end of the ground
shield, the power density reaches 500 kW/m3
. To put these
power densities in context, the volumetric heat capacity of all
dense solids is about 2.5 × 106
J/m3
-K so that a power density
of 42 kW/m3
implies an adiabatic temperature rise of 0.017 K/s,
which is easily accommodated through thermal diffusion. A
power density of 500 kW/m3
implies an adiabatic temperature
rise of about an order of magnitude greater or about 0.2 K/s,
which can probably be accommodated given the proximity of
the metal corona shield as a heat sink and the relatively small
volume over which this high power density occurs. In practice, a
time harmonic finite element computation with coupled thermal
and electric fields and appropriate boundary conditions would
provide a good estimate of the steady state temperature, even for
field-dependent electrical and thermal properties.