1. Frequency Dependence: Dielectric Constant and Dielectric Loss
Lecture 3
Frequency Dependence: Dielectric Constant and Dielectric Loss
Md Arafat Hossain
Department of Electrical and Electronic Engineering, Khulna University of Engineering and Technology, Khulna, BANGLADESH

2. Outlines Frequency Dependence: Dielectric Constant and Dielectric Loss
Debye Equations, Cole-Cole Plots, and Equivalent Series Circuit

3. Dielectric Loss (Cont.)
Background:
Dielectric medium under alternating field
Polarization will be different than the static case
Example: orientational (dipolar) polarization,
Varying field changes magnitude and direction continuously, and it tries to line up the dipoles one way and then the other way and so on
If the instantaneous induced dipole moment p per molecule can instantaneously follow the field variations, then at any instant + - p0 E
and the polarizability aj has its expected maximum value from dc conditions, that is,

4. Dielectric Loss (Cont.)
There are two factors opposing the immediate alignment of the dipoles with the field.
Thermal agitation that tries to randomize the dipole orientations
Viscosity of the medium: the molecules rotate in a viscous medium by virtue of their interactions with neighbors, which is particularly strong in the liquid and solid states and means that the dipoles cannot respond instantaneously to the changes in the applied field.
But if E changes rapidly (f - high),
dipoles cannot follow the field and, as a consequence, remain randomly
oriented. Therefore αd will be zero….so E can’t induce a dipole.
But at low f, the dipoles can respond rapidly to follow the field and αd has its maximum value.
We need to find the behavior of αd as a function of f co so that we can determine the dielectric constant er by the Clausius-Mossotti equation.
(2)

5. Dielectric Loss (Cont.)
Frequency dependence of dielectric constant
After a prolonged application, corresponding to dc conditions, the applied field across the dipolar gaseous medium is suddenly decreased from E0 to E at a time we define as zero,
So the induced dipole moment per
molecule has to decrease, or relax, from
αd (0)E0 …………..αd (0)E.
In a gas medium the molecules would be moving around randomly and their collisions with each other and the walls of the container randomize the induced dipole per molecule. Thus the decrease, or the relaxation process, in the induced dipole moment is achieved by random collisions.
Assuming that τ is the average time, called the relaxation time, between molecular
collisions, then this is the mean time it takes per molecule to randomize the
induced dipole moment.

6. Dielectric Loss (Cont.)
If p is the instantaneous induced dipole moment, then [p -αd (0)E] is the excess dipole moment, which must eventually disappear to zero through random collisions as t → ∞
It would take an average τ seconds to eliminate the excess dipole moment
The rate at which the induced dipole moment is changing is then
…………..(01)
The –ve sign indicates a decrease
The above equation can be used to obtain the dipolar polarizibility under ac condition.
In engineering form,
For an ac field,
From eq. (01)
…………..(02)

7. Dielectric Loss (Cont.)
Solution of Eq.(02)
…..(03)
…..(04)
Where,
Eq. (04) represents the orientational polarizability under ac field conditions

8. Dielectric Loss (Cont.): f-dependence
We can easily obtain the dielectric constant εr from αd (ω) by using Eq. (blue box), which then leads to a complex number for εr since αd itself is a complex number. By convention, we generally write the complex dielectric constant as
…..(05)
(04)

9. Dielectric Loss (Cont.): f-dependence
Consider the capacitor, which has this dielectric medium between the plates. Then the admittance Y, i.e., the reciprocal of impedance of this capacitor, with εr given in Eq. (05) is
…..(06)
Which can be written as
…..(07)
…..(08)
Where
…..(09)

10. Dielectric Loss (Cont.): f-dependence
Thus the dielectric medium behaves as if C0 and Rp were in parallel. There is no real electric power dissipated in C, but there is indeed real power dissipated in Rp because

11. Dielectric Loss (Cont.): f-dependence
In engineering applications of dielectrics in capacitors, we would
like to minimize εr " for a given We define the relative magnitude of e εr' with respect to ef r through a quantity, tan 8, called the loss tangent (or loss factor), as

13. Lecture 5: Debye Equations, Cole-Cole Plots,
and Equivalent Series Circuit
Debye Equations
Consider a dipolar dielectric in which there are both orientational and electronic polarizations, αd and αe respectively, contributing to the overall polarizability.
At high f, orientational polarization will be too sluggish too respond, αd = 0, and the εr will be εr∞. The dielectric constant and polarizabilities are generally related
through
Independent of f over the typical frequency range of operation of a dipolar dielectric, well below optical frequencies.
…………..(01)
Relative permittivity at high f, where orientation polarization is negligible

14. Debye Equations, Cole-Cole Plots, and Equivalent Series Circuit
Writing εr in terms of real and imaginary parts, and f-dependence αd (ω)
…………..(02)
Contribution of orientational polarization to the static dielectric constant εrdc, i.e (dc condition – constant value/f-independent value that presents always)
Simplify Eq. (02)…...equate real and imaginary parts to obtain what are known as Debye equations :
Debye equations

15. Debye Equations, Cole-Cole Plots, and Equivalent Series Circuit
Hypothetic and reality:
The imaginary part εr " that represents the dielectric loss exhibits a peak at ω = 1 /τ which is called a Debye loss peak.
Many dipolar gases and some liquids with dipolar molecules exhibit this type of behavior
In solids, the peak is typically much broader because we cannot represent the losses in terms of just one single well-defined relaxation time but a distribution of relaxation times arises from different molecular process.
It was assumed that, the dipoles do not influence each other either through their electric fields or through their interactions with the lattice. In solids, the dipoles can also couple, which complicates the relaxation process. Nonetheless, there are also many solids whose dielectric relaxation can be approximated by a nearly Debye relaxation or by slightly modifying it.

16. Debye Equations, Cole-Cole Plots, and Equivalent Series Circuit
In dielectric studies of materials, it is quite common to find a plot of the imaginary part (εr") versus the real part (εr') as a function of frequency (ω). Such plots are called Cole-Cole plots after their originators.
While for certain substances, such as gases and some liquids, the Cole-Cole plots do indeed generate a semicircle, for many dielectrics, the curve is typically flattened and asymmetric, and not a semicircle

17. Debye Equations, Cole-Cole Plots, and Equivalent Series Circuit
A capacitor with a dipolar dielectric and its equivalent circuit in terms of an ideal Debye relaxation.
High f component
Typical f component
Notice that in this circuit model, Rs, Cs, and C∞ do not depend on the frequency, which is only true for an ideal Debye dielectric, that with a single relaxation time r.