# Dielectric constants and conductivity

Discussion in 'General Electronics Chat' started by rperea, Apr 13, 2009.

1. ### rperea Thread Starter Member

Apr 7, 2009
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Is there a simple relation between dielectric constants and conductivity one? I know conductivity is the reciprocal of resistivity but what about dielectrics. Also, I am aware that dielectric constants are use for insulator while conductivity for conductors but still I would like to know any relation.
Regards.

2. ### Ron H AAC Fanatic!

Apr 14, 2005
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There is no relationship between the conductivity and dielectric constant of a material. The capacitance between two conductors will be proportional to the dielectric constant of the material between the conductors, assuming the spacing and area are fixed.

3. ### Dave Retired Moderator

Nov 17, 2003
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In dielectrics, the permittivity can be expressed as a complex number:

ε = ε' - jε''

ε' is the dielectric constant

ε'' is the dielectric loss factor

The dielectric loss factor is a function of conductivity and frequency:

ε'' = σ/(2∏f)

Where: ω = 2∏f

So the permittivity can be expressed as a function of the dielectric constant and the conductivity:

ε = ε' - jσ/ω

The concept of conductivity in a dielectric manifests itself in a description of how lossy the dielectric material is - that how well does the dielectric material convert the energy in the E-field at a certain frequency into heat; in fact this is a fundamental concept underpinning how a microwave oven cooks food.

Dave

4. ### steveb Senior Member

Jul 3, 2008
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Adding to the ideas above. RonH is talking about the general case, for which there is no relationship between those values. Dave is talking about the complex permitivity and how it is related to the dielectric constant and the conductivity. This concept is generally used for linear media.

In the restricted case of linear media, there are the Kramers-Kronig relations which expresses conductivity as an integral of dielectric constant over all frequencies, and also expresses dielectric constant as an integral of conductivity over all frequencies.

The following reference gives the basic expressions, but you will need a good reference to come to grips with the concepts since it is not really a very simple relationship (which was what you were asking for).

http://scienceworld.wolfram.com/physics/Kronig-KramersRelations.html

5. ### Dave Retired Moderator

Nov 17, 2003
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Thanks for that Steve, I forgot about the Kronig-Kramers relationship for dielectrics, however it must be stressed this is certainly not a trivial treatise of the relationship (see the Gorges paper to see what I mean).

An interesting thing to note about the mathematical treatise I gave above is that it seems to corroborate Ron's point that the dielectric constant and conductivity are independent entities; from a strict circuits perspective this is the correct way to view this. However, if we look at the dielectric constant and conductivity as a function of another variable it illustrates that there is infact a subtle link between the two.

For example, consider the dielectric constant and conductivity as a function of frequency which is characterised in first-order case by the Debye Relaxation Equation and also the Havriliak-Negami Relaxation Equation which takes account of the inherent asymmetry and broadness of the dielectric dispersion curve. As the material approaches a relaxation point, the dielectric loss factor (and hence conductivity) increases - this corresponds to a drop in the dielectric constant. What is actually happening here is that the material is becoming less of an energy store and more of an energy dissipater, something which in itself is related to the polarisation characteristics of the material as a function of the applied frequency.

Each material is different and there are many more factors than that discussed here, temperature, density, compositional changes in the material, to name a few.

I recommend the Böttcher and Bordewijk text if anyone is interested - it is heavy going, but is to my mind the standard text for those wanting to understand electric polarisation and dielectrics.

Dave

6. ### studiot AAC Fanatic!

Nov 9, 2007
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Interesting chat folks.

I wonder what application the OP had in mind?

Electromagnetic wave transmission theory?

Or perhaps properties of ionic solutions in physical or analytical chemistry?

The question has meaning in both of these fields and probably others too.

For a material there is a dimensionless number, $\frac{\sigma}{\omega\epsilon}$ which allows us to distinguish between dielectrics, quasi-conductors and conductors.

σ is the conductivity, ω is frequency and ε is permittivity

At 1khz the earth acts as a conductor, at 10Mhz as a quasi-conductor and at 30Ghz as a dielectric

Acknowledgements to 'thatoneguy' for the instruction in TEX - It works yay!

Then of course we have the related phenomenon of polarisation so that

Electric flux density D = ε0E + Polarisation ,P

Vectors are shown in bold.

One thing to remember is that different units (dimensions)
of conductivity and dielectric constant.

7. ### Ron H AAC Fanatic!

Apr 14, 2005
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You guys are way over my head. Does ESR have any effect on the dielectric loss factor?

8. ### Dave Retired Moderator

Nov 17, 2003
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Ron, what is ESR? Electron Spin Resonance?

Dave

9. ### steveb Senior Member

Jul 3, 2008
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I think you can say that dielectric loss has an effect on ESR - equivalent series resistance. The following reference gives a decent explanation.

http://www.lowesr.com/QT_LowESR.pdf

Last edited: Apr 14, 2009
10. ### Dave Retired Moderator

Nov 17, 2003
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Damn me! How did I not get "Equivalent Series Resistance" - I've must have Tomographer's head on!

That link does a good job of illustrating that, yes, it does have an effect on ESR; naturally to varying degrees dependant on the frequency range of operation and the lossyness. In some applications, although it is a factor, its effects are swamped by other physical factors of the capacitor which contribute to a greater extent to the ESR.

Dave

11. ### Ron H AAC Fanatic!

Apr 14, 2005
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The equivalent circuits in Fig. 2 work for a single frequency (ω), but they obviously doesn't work for DC, impulses, steps, or anything other than ω. I don't understand why they developed that model. I can't see any relevance to circuits that are affected by ESR, such as power supplies. A crappy dielectric can have a lot of ESR, and still be a nearly perfect insulator. I would guess that there are also leaky dielectrics that have very low ESR.
Does that shed any new light on my question about whether ESR can affect the dielectric loss factor? In other words, can a perfect insulator still have a poor dielectric loss factor, due to ESR?

12. ### Dave Retired Moderator

Nov 17, 2003
6,960
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Correct. That is because of the mechanism by which the pure dielectric loss factor of a dielectric is defined - it arises primarily due to polarisation effects on the molecular level of the dielectric; think how a molecular dipole reacts (realigns) to the application of an alternating electric-field and how that changes over a frequency sweep. At DC it is a concept that doesn't fundamentally work because the lossyness is a function of the dipole dissipating energy as it rotates in-line with the applied field.

To complicate things further it isn't just due to polarisation effects, but also atomic, electronic and Maxwell-Wagner interfacial losses. When you consider dielectric loss including all these potential loss mechanisms, including electric conduction which would be one of the main mechanisms on typical dielectrics in commercial capacitors, the lossyness is not merely a function of energy dissipation arising from molecular rotation within the dielectric, but also I^2R losses. The complex notion for permittivity - as I stated before - comprising the dielectric constant and dielectric loss factor, typically lumps all aspects of the conductivity into the effective dielectric loss factor. Why? For high-frequency applications, it gives a "good enough" characterisation of the conductivity at a frequency in light of the variability of dielectric loss as a function of temperature, physical/chemical composition, density, etc. As you can probably see, at low-frequencies it becomes quite vague as the conduction mechanism is more prominent than the polarisation mechanism.

The effective dielectric loss factor is actually characterised by 4 main components:

ε''(effective) = ε''(dipole) + ε''(electronic) + ε''(atomic) + ε''(Maxwell-Wagner)

Where:

ε''(dipole) is the true dielectric loss factor resulting from polarisation effects.

And:

ε''(electronic) + ε''(atomic) + ε''(Maxwell-Wagner) = σ/ωε0

Where σ here is conductivity in traditional sense.

Kinda returning to the original points in this thread, the important point to note is that on a macro-scale, the components of the effective dielectric loss all manifest themselves in the same way - dissipation of energy; it is very difficult to disassociate the components, therefore if you model them as a lumped conductivity you can characterise the lossyness of the material with high-accuracy (at higher frequencies at least).

The ESR in capacitors is, to varying degrees, a function of the dielectric loss factor. The problem in asking this question is, how do we define perfect insulator? Whilst in the presence of an applied-field there may be no free electrons in the conventional sense of electrons in a metal wire which would give rise to current in the traditional sense resulting in Joule Losses, the polarisation lag on the molecular level is infact taking energy from the applied-field (it has pure dielectric loss and an equivalent conductivity).

Even though I gave the referenced article a once over I haven't had chance to give it a full going over, so I may be hitting around some of the points you are making with respect to the article and ESR. That said, I must admit to understanding dielectrics more in the physics/EM sense rather than the circuits sense.

Dave

Last edited: Apr 14, 2009
13. ### mbohuntr Active Member

Apr 6, 2009
413
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Sorry to butt in here, but do you think he means capacative reactance?(Xc)

14. ### Ron H AAC Fanatic!

Apr 14, 2005
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I could be wrong, but my best guess is that our OP was talking about the conductivity that you can measure with an ohmmeter. I'll bet the physics discussions went right over his head, like they did mine. If he comes back with a big thanks for all the answers, I will be surprised.
I gave a reply based on my experience as a circuit designer, and I'm still not sure whether I was right or wrong.

15. ### mbohuntr Active Member

Apr 6, 2009
413
32
OK, I was thinking of total ESR based on frequency. OT, I just finished researching a paper on circuit design... I'm sorry for your frequently frustrating occupation....

16. ### steveb Senior Member

Jul 3, 2008
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I agree. The original question was not clear on exactly what the application is. There was also no indication of his background to indicate what level of detail should be provided. I mentioned the Kramers-Kronig relations because of his last statement.

Last edited: Apr 14, 2009
17. ### Dave Retired Moderator

Nov 17, 2003
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The OP could very well have meant the conductivity when measured with an ohmmeter. If you ask me, I think a thorough answer from a range of perspectives has been given - note I did say that your original answer "from a strict circuits perspective [was the] correct way to view this". That said, in a strict physics sense the dielectric constant and loss factors are quite complex concepts. The OP is free to interpret and probe all and any posts as necessary.

Might be an idea to create a thread-fork of all the physics related discussion if anyone wants to discuss these points further.

Dave

18. ### KL7AJ Senior Member

Nov 4, 2008
2,013
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The important factor, that this is a COMPLEX number, has been appropriately emphasized above. The conductivity of the Earth, used for radio propagation purposes, is always expressed as a complex number, as is the ionosphere itself.

In any physical system, the real component represents loss or dissipation of energy, while the imaginary component represents energy storage (often manifesting itself as phase shift or change of velocity)

Eric

19. ### studiot AAC Fanatic!

Nov 9, 2007
5,005
513
Your LHS above has dimensions F/m and the RHS is dimensionless?

See my comment on units in post #6.