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# Time Domain Measurement

The purpose is to measure the permittivity ε(t) and conductivity σ(t) functions  of a sample material in dependence of time t. The material may be solid or liquid and is usually placed between two electrodes to form a capacitor.

At time t=0, voltage step from 0 V to U0 is applied to the sample capacitor. The response current I(t) is measured as a function of time. The time dependent capacity is calculated from

$C(t)=\frac{Q(t)}{U_0}=\frac{1}{U_0}\int_0^t I(t')dt'$

It is directly related to the time dependent permittivity function by

$\epsilon(t)=\frac{C(t)}{C_0}$

where C0 represents the capacity of the empty capacitor without any sample material in between. If the electrodes are arranged in parallel, this empty-cell capacity will be given by

$C_0=\epsilon_0 \frac A d$

where A denotes the area of one electrode, d the spacing between the electrodes and ε0=8.854*10-12 As/Vm is the vacuum permittivity.

It should be noted that ε(t), in the general case, comprises both a permittivity contribution (reflecting the reorientation of molecular dipoles) and a conductivity contribution due to mobile charge carriers. The same applies to the time dependent conductivity function which is another representation of ε(t):

$\sigma(t)=\epsilon_0\left( \frac {d\,\epsilon(t)}{dt}-\delta(t) \right)=\frac{I(t)}{U_0A}-\epsilon_0\delta(t)$

The delta function δ(t) describes the current peak of the empty electrodes (vacuum permittivity) at t=0, resulting in ε(t) = 1 and σ(t) = 0 for the vacuum. Refer to Dielectric Spectroscopy and Impedance Spectroscopy of Materials for details and typical properties of ε(t) and σ(t).