### 1- Phasor mode

In Phasor mode the dielectric constant is considered to be a complex number consisting:

ε = ε_{r}ε_{o} + j * σ/ω

where:

ε_{r} = relative permittivity

ε_{o} = permittivity of free space

σ = electrical conductivity

ω = angular frequency = 2πf

In most cases, for most materials, either the real or the imaginary part will be orders of magnitude larger than the other at the frequency of operation. If it is the real part that is much larger the material for practical purposes behaves as, and should be modeled as, a perfect insulator. If it is the imaginary part that is much larger, the material is operating as a conductor.

For example, a typical “insulator” used in an electric model might have a conductivity of 10-12 /Ω-m and a relative permittivity of 4. Using INTEGRATED’s Quick Engineering Tool for Lossy Dielectrics, this produces an imaginary part of the relative permittivity of 0.0003 when the frequency is 60 Hz. In this case it is very accurate to model with the conductivity=0 for faster results.

### 2- Transient

In the general time dependent case, the response of a model to time varying signal involves movement of charge and changing of polarization. These give rise to the characteristic time responses at various points in the system. In the case of a harmonic signal this can be treated at real and imaginary parts (above), but in general it is simply considered a “transient response”. Consider the very simple parallel plate system shown below:

If we monitor the voltage at the interface over time, it will behave the same as this circuit:

The dielectric layer behaves as a resistance in parallel with a capacitance, and the empty space layer behaves as a capacitance. The interface is designated by a node labeled “i”. The conductivity was selected so that the INTEGRATED Quick Engineering Tool for Lossy Dielectrics would predict a “time constant” of 0.2 s. An analysis with ELECTRO then determined the following voltage response at the interface:

Note:

- Since the capacitance of the dielectric layer is 3x that of the empty layer, 3/4 of the voltage is across the empty layer when the voltage is first turned on.
- Once the voltage is turned on, charge begins to leak through the material. Over time the voltage across the dielectric is decreased.
- In one time constant (0.2 s) the voltage across the dielectric layer is decreased by a little more than half, but not as much as 1/e. The latter would be the response if the system consisted entirely of the dielectric layer leaking charge. In the whole model (“circuit”) the response will be different, but of a similar magnitude. Hence the time constant estimation given INTEGRATED’ Quick Engineering Tool for Lossy Dielectrics provides the relevant time frames for setting up transient parameters.

### 3- Static

Referring to the simple case in section 2 (Transient), a “static” model is actually a transient one of two limits after a DC voltage is turned on: either it is the initial response over times much shorter than the time constant or it is the final state after all transient effects have decayed away. As an example, check the calculator for conductivity=10-15 S/m and relative permittivity=4. The time constant is approximately 10 hours. So if the voltage is turning on an off over a few seconds or minutes, the conductivity is irrelevant. However, if it is turned on and left on for months, the long-term behavior is dominated by the conductivity of the material.

- In the high voltage industry a failure is often found days or weeks after turning equipment on. Due to the high resistivity (low conductivity) of the materials used the time constant may be very long, but the equipment may also be ON continuously for a long time. In this case two analyses are needed for potential high field problems in the materials: a permittivity mode solution for the short term response, and a conductivity mode solution for the long term response. If the transitional states are of interest, those need to be modeled in transient mode.