Data analysis after dielectric characterization often requires modeling of dielectrics which is helped by their representation as equivalent electrical circuits. Incidentally, a perfect dielectric material can be modeled by an equivalent RC parallel circuit as shown below. 

Let us consider the admittance (Y) of the material representing the above circuit which is inverse of the impedance (Z), and is expressed as

(4.108)

where and

(4.109)

Hence admittance now is  

(4.110)

 
Now if we consider time constant for the segment R₂C₂ as τ₂ = R₂C₂, then admittance can be written as

		(4.111)

Now, since admittance can be related to the dielectric constant as

(4.112)

This gives

and

(4.113)

which have the same form as Debye equations with C₂ = (ε_s - ε_∞) C₀ and C₁ = ε_∞.

In 1941, K.S. Cole and R.H. Cole explained the behavior of dielectrics in alternating fields in which they plotted the electrical response of dielectric materials to the alternating fields as a function of frequency.
By this technique, they were able to identify and relate the observed relaxation effect with the atomic and microstructural features of the materials.

However, Cole and Cole used modified Debye equation which is

(4.114)

where α is a parameter which describes the distribution of relaxation times in the material.

For an ideal dielectric with well defined relaxation time i.e. α = 0, the system would be represented by the Debye equations i.e. equations (4.104-4.106).

As a result, if we plot  ε_r^'' (ω) vs ε_r^' (ω)   i.e. imaginary vs real part of the dielectric constant on a complex plane, we get a semi-circle.