# 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 *U _{0}* 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

It is directly related to the time dependent permittivity function by

\[\epsilon(t)=\frac{C(t)}{C_0} \]where *C _{0}* 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

where *A* denotes the area of one electrode,
*d* the spacing between the electrodes and *ε*_{0}=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):

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)*.