November/December 2004 — Vol. 20, No. 6
F E A T U R E
A R T I C L E
dielectrics are nonlinear if subjected to sufficiently
high electric fields.
For example, Figure 1 shows fits
to data for the conductivity of two cable dielectrics, one with an
activation energy of 0.56 eV and the other with an activation
energy of 0.98 eV.
The former would be appropriate for a DC
cable, while the latter would be typical of an AC cable insulation.
At low fields, charge carriers are detrapped thermally, and the
field simply makes hopping more likely in the direction of the
field than against the field.
However, as energy is gained from
the field as a charge carrier travels from one trap to the next, it
becomes comparable to the thermal energy; the energy gained
from the field becomes a significant factor in detrapping, which
results in the conductivity becoming an exponential function of
The thermal energy at room temperature is about 0.025
eV, and the typical distance between traps is about 2.8 nm.
the energy gained from the field in moving between traps be-
comes equal to the thermal energy at a field of about 9 kV/mm,
which is in good agreement with the transition from constant to
exponentially increasing conductivity seen in Figure 1.
While hopping theory predicts a hyperbolic sine relationship
between the current density and electric field, i.e., constant con-
ductivity at low fields with a transition to an exponential increase
in conductivity with field at higher fields, the actual low field
conductivity can be more complex, as seen in Figure 2.
four dielectrics shown, only EPR2 (ethylene propylene rubber)
and biaxially oriented polypropylene capacitor film (BOPP) fit
the model reasonably well. The conductivity of (degassed)
TRXLPE (tree retardant crosslinked polyethylene) has a large
field-dependence in the low field region, and EPR1 has a rela-
tively small field dependence.
Nonlinear dielectric properties can be used to control the elec-
tric field as in a distribution cable termination that must operate
under both AC and impulse conditions.
Under AC conditions,
capacitive grading is likely to be employed; however, under im-
pulse conditions, where the voltage can be nearly 10 times greater,
capacitive grading may not be adequate, and nonlinear grading
On the other hand, nonlinear resistive grad-
ing may dissipate energy at levels not acceptable under normal
operating conditions but that are not problematic during the few
tens of microseconds of an impulse.
In other situations, nonlin-
ear conduction “overcomes” the large temperature dependence
in the conductivity that would otherwise cause highly nonlinear
grading under DC operating conditions.
This occurs in DC cables
as well as in ZnO arrester elements when heat sinks are present
between the elements, which results in an axial temperature gra-
dient under impulse-current conditions.