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# Calculation of Relative Dielectric Constant of Piezoelectric?

11/10/2017 9:31 AM

Hi everybody

Finite element analysis of Cymbal piezoelectric transducers for harvesting energy from asphalt pavement

In this paper, The governing equations of piezoelectric is:

Where tp=2mm,Diameter(D)=32mm and piezoelectric material is PZT 5H.

As far as I know ε0= 8.854 pF/m.

in paper relative dielectric constant at constant strain is:

From simulation in abaqus i get the V=96.98

they mentioned In paper:

The electric potential is about 97.33V of the design Cymbal. 0.06 J electric energy can be storage in that Cymbal.

Then U is: 0.06 J

Now my main question is how can i calculate the εTr ?

Because U is the final answer. and when i put all variable in eq 8. and pick Dielectric Constants, @ 1KHz, 3500 from this site (for material: PZT-5H):

https://www.steminc.com/PZT/en/piezo-materials-properties

My U will get this:

U=5.9024e-05

As you can see There is a lot of difference.

By the way if i put relative dielectric constant at constant strain matrice in eq 8. how can this possible , one side is scalar and other side is matrices ?

With best regards.

yassou.

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#1

### Re: Calculation of Relative Dielectric Constant of Piezoelectric?

11/10/2017 9:55 AM

I think a more important question to ponder is, will generating electricity from pavement, given the massive infrastructure modifications required, ever be economically feasible?

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#5

### Re: Calculation of Relative Dielectric Constant of Piezoelectric?

11/10/2017 5:33 PM

Yep, just add up an engine and generator, then it is set to go.

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#6

### Re: Calculation of Relative Dielectric Constant of Piezoelectric?

11/11/2017 4:06 AM

Who can answer that? We can get some guesses (based on our background).

But as far as I know, taking the first steps take more energy. Now, it has to worse Take those steps or not, I cannot say that.

Wisdom is how far you can see a depth of field.

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#2

### Re: Calculation of Relative Dielectric Constant of Piezoelectric?

11/10/2017 10:48 AM

By the way if i put relative dielectric constant at constant strain matrice in eq 8. how can this possible , one side is scalar and other side is matrices ?

The dielectric constant is a matrix because the material is non-isotropic. This means that the dielectric constant varies depending on the direction it is measured. So you would have a scalar value for dielectric constant when you take into account the orientation.

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#7

### Re: Calculation of Relative Dielectric Constant of Piezoelectric?

11/11/2017 4:14 AM

Thanks for Reply, I know that, but as far as I know εTr is rate type scalar (means it has no unite), from other hand relative dielectric constant at constant strain unit is (Farad/meter).

Abaqus documentation:

And U (Stored electrical energy) unit is Joule.

You can see the difference in below:

And This:

And also if I pick εT33r as a εTr (num 9 matrices,1700) for the calculation. At the end answer for the U will not be the 0.06 J.

I took Abaqus εTr matrice:

Maybe I make some mistake in understanding of that concept.

With best regards.

Yassou.

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#11

### Re: Calculation of Relative Dielectric Constant of Piezoelectric?

11/11/2017 10:00 AM

Scalar doesn't mean unitless, it means a single magnitude, for example, temperature. A vector has more than one quantity, for example, velocity or electric field. A matrix or tensor is a two-dimensional array of numbers, for example, stress or strain. Any one of these types could be unitless or have some units associated with them.

εTr is a unitless matrix. The subscript r means "relative to e0", and it will always be multiplied by e0 with units (F/m) to get the true dielectric value, εT, which will be a matrix with units (F/m).

I hope this helps.

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#3

### Re: Calculation of Relative Dielectric Constant of Piezoelectric?

11/10/2017 12:28 PM
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#8

### Re: Calculation of Relative Dielectric Constant of Piezoelectric?

11/11/2017 4:16 AM

Thanks for Reply, I look at your link, sorry but I cannot catch what you want to mentioned.

Can you specify that.

With best Regards.

Yassou.

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#13

### Re: Calculation of Relative Dielectric Constant of Piezoelectric?

11/11/2017 1:56 PM

I was thinking there may be some examples of calculations in this paper that you may refer to....

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#4

### Re: Calculation of Relative Dielectric Constant of Piezoelectric?

11/10/2017 1:45 PM

Now my main question is how can I calculate the εTr ?

Some background: If you have two parallel conducting plates with opposite charges, there is an electric field between them. The strength of the electric field is proportional to the amount of charge, inversely proportional to the distance between them. The constant of proportionality is 1/e0.

Insert a polarizable material, a dielectric. The electric field separates the charged ions in the lattice (dielectric displacement), thereby reducing the electric field. This results in the permittivity being effectively larger, so the resulting field is proportional to 1/ereo. (er > 1)

A piezoelectric material changes dimensions when an electric field is applied. Whether it is allowed to do this (constant stress) or constrained from changing dimension (constant strain) determines the permittivity or how much the ions are allowed to move in response to the field. Permittivity with constant stress (eT) and with constant stain (eS) are different and are dependent on the directions of the displacement and the electric field, which is why they are represented by matrices.

Back to your question, as far as I know, eT and eS are each determined by measurement from capacitance measurements with the piezoelectric material unclamped and clamped respectively.

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#9

### Re: Calculation of Relative Dielectric Constant of Piezoelectric?

11/11/2017 4:17 AM

Thanks for Reply, That means if you have εSr you cannot get the εTr ?!

Am I correct, There is no calculation between them.

Then in this paper for PZT-5H, i have εSr , and I cannot get the εTr.

It seems I stuck. Because formula (Eq.8) want εTr and I don't have it.

With best Regards.

Yassou.

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#10

### Re: Calculation of Relative Dielectric Constant of Piezoelectric?

11/11/2017 4:36 AM

Guys, by considering εTr as Unknown in matlab software.

It give me this value:

εTr=3.5579e+06 .

Look Below:

https://imgur.com/a/Tns4e

To me is like every thing is get collapse, nothing match together?

Is there any body know what will be happened ?

εTr from Matlab is not near to any values from references ?!

For examples:

https://www.steminc.com/PZT/en/piezo-materials-properties

or calculation i did ?

With best regards.

yassou.

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#12

### Re: Calculation of Relative Dielectric Constant of Piezoelectric?

11/11/2017 10:09 AM

Hello again guys.

it seams i get near of the answer.

First, look below:

So, ε33=1.3020e-08, and, ε33r=1470.

in abaqus software, as paper, the model is in Axisymmetric type:

thickness of PZT (tp=2mm),electric potential (voltage) caused by polarization changing (V=97.33).

Now surface area of PZT disk (A):

in model dc=32mm then rc=16.

then A=pi*(r^2)=804.2477 (mm^2) or 8.0425e-04 (m^2)

In eq.8 all variable are known:

So:

with substituting in equation:

U in the paper is:

0.06 mJ

my U is:

0.0247902 mJ

so it seams it's technical problem...

can anyone guide me ?

with best regards.

yassou.

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#14

### Re: Calculation of Relative Dielectric Constant of Piezoelectric?

11/11/2017 9:26 PM
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