The dependency of electric field distribution on time and temperature is obvious in HVDC system. This is due to the resistivity/conductivity dependency on temperature and electric field [1, 44]. It is also clear that the formation of space charge within the insulation material is depended on both temperature and external applied field [15]. Therefore, the dependent conductivity has a great role on space charge and then on field distribution in DC system. Under steady state DC condition the electric

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Moreover, after applying the voltage for a long time, if there is no space charge built up, the current density gain the steady state value, and equation 3.2 and 3.4 become:

∇.J=∇.(σE)=0 (3.6)

Based on equation 3.6, the field distribution is depended on conductivity variation inside the material [1, 45, 46]. Furthermore, according to equations 3.5 and 3.6, space charge can be accumulated in insulation material in steady state condition due to the variation of permittivity to conductivity. As a result, by combining equation 3.2, 3.3 and 3.6, the equation of determining space charge in steady state condition can be written

∇.(ε_0 ε_r E)=∇.(ε_0 ε_r J/σ)=J.∇((ε_0 ε_r)/σ)+ (ε_0 ε_r)/σ ∇.J=J.∇ ((ε_0 ε_r)/σ)= ρ (3.7) or ρ=σE.∇( (ε_0 ε_r)/σ)

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Based on this, when the time derivation of charge density goes to zero, field distribution and space charge obtained the steady state condition, hence equation 3.9 reduces to 3.8. Furthermore, when the conductivity increases, the dielectric time constant decrease, thus the field distribution can reach the steady state condition rapidly.

The three equations 3.2, 3.3 and 3.4 demonstrate the electrical behaviour of a weakly conductive of insulation material, while equation 3.9 which is the result of combining these three equations describe the accumulated space charge when the time derivation of current density is not zero and the ratio between permittivity and conductivity ( (ε_"0" ε_"r" )/σ) is not constant across the insulation. The permittivity of insulation material is almost independent of temperature and electric field, hence it can be considered as a constant [29]. However, the electrical conductivity of the dielectrics is strongly dependence on both temperature and electric field