4.10 Impedance Spectroscopy

The frequency response of the impedance of above circuit would also yields a semi-circle in the complex plot between real and imaginary parts of the dielectric constant and such a response represent a capacitor with losses. Such technique of characterizing dielectrics is called Impedance Spectroscopy.

The intercepts of the semi-circle on the x-axis represent high and low frequency dielectric constants or ε_r∞' and ε_rs'  respectively.

The maxima of the semi-circle occurs at

(4.115)

The semicircle equation turns out to be

( - ) + =

which can be obtained by removing ωτ from the equations (4.105a) and (4.105b). 

Alternatively in a semi-circle one can draw two vectors, X and Y which are always perpendicular to each other and are related as y = -iωτ.xThis relation can easily be obtained just by using the Debye’s equation (4.98) and simple vector additions and subtractions.

Basically any point on the surface of the semi-circle of diameter (ε_rs' - ε_r∞' ) in this complex plane can be defined by these vectors.

In the figure above, a perfect semicircle represents a narrow distribution of relaxation times indicating only one primary mechanism of polarization.

However, if there is the presence of a tail on the low frequency side, as shown by dashed line i.e. increasing ε_r'' , indicates large distribution of relaxation times and usually is due to high losses in the material. 

In case a material has more than one contribution to the impedance, which is often the case with polycrystalline ceramics where grain, grain boundaries and electrode-ceramic interface have distinct contributions, then one can witness more than one semi-circle, often overlapping each other. An idealized picture of such a scenario can be see below. One of the ways to model such a behavior may be using three series-parallel RC elements circuit as shown below.