Calculation behind Fringing Field Sensor

[tommyers]

Hi,

Could someone explain the elementray concept and calculations behind a fringing field sensor. As shown:

http://www.ee.washington.edu/research/seal/projects/moisture sensing/project overview.htm#how"

Taking the elementray equation behind the parallel plate arrangement to be,

C = (Eo.Er.A)/d, how does this adapt for the fringing field arrangement shown?

What is the underlying concept behind these i.e. what cause such large fringing?

Regards

Tom

 

[lalbatros]

Hello Tom,

When you first mentioned "fringing" here on the PF, I did not really understand the exact meaning, and I translated as "edge fields".

On the site you give, the "fringing sensor" is clearly illustrated in the first figure. The geometry of the system is somawhat different than for a traditional disk capacitor, but it doesn't change anything to the the relation Q = C V . Of course, the calculation of the capacity might be a little bit more complicated.

Considering figure (c), for example, you can stil write C = e A / d , but you need to replace d by the average length of a field line. The length you can evaluate approximatively by taking the field line starting in the middle of the plate and asuming it is close to a circle (to be checked !).

The "Multi-wavelength sensor" is also interresting. The figures on the web page does not indicate how the different electrodes are used. One way is to connect together the even plates and the odd plates and sense the capacity between these two set of plates. It this case, the formula C = eA/D would still apply and easily used for an approximation. There are of course numerous ways to connect such a multi sensor and this might allow many games! Note also different connections might be used at the same time using different frequencies to probes the respective capacities. Quite funny!

Concerning the fringing, it is determined by the basic laws of physics, electrostatics. The field has to go from the positive plate(s) to the negative plate(s). Without charge, away from the plates, the field is divergence free since (div E = 0), which means that flux is conserved in flux tubes, and which results in this fringing.

Michel

 

[tommyers]

Concerning the fringing, it is determined by the basic laws of physics, electrostatics. The field has to go from the positive plate(s) to the negative plate(s). Without charge, away from the plates, the field is divergence free since (div E = 0), which means that flux is conserved in flux tubes, and which results in this fringing.

Could you elaborate or give me some background on this?

Regards

Tom

 

[lalbatros]

Tom,

For a background, you should take a book on electrostatics (or electromagnetism).
I would recommend Feynmann, but there are many others.
Maybe you have read some already.
What have you read already on electrostatics ?

To elaborate, it would be better that you ask more precise questions, otherwise we would have to answer with a whole book.

Now, to be more concrete, fringing is quite a simple thing.
Take the electric field from a point charge: you cannot find an electric field with more fringing: it goes equally in all directions. Point charges produce a (widely) fringing field.

For a two disk capacity, you could start with a rough approximation: uniform charge densities. Each point for a plate has a charge producing an electric field in all directions. The fringing from the two-disk arrangement results from the (widely) fringing field from each point from each disk. Remember that the field is proportial to the charge and that the field from several charges can be summed up (as vectors for the electric field of course and as numbers for the potential).

Actually, you should not ask this question: fringing, fields going in many directions, is normal.
You should rather ask the opposite question: when and why and how can an electric field be confined and have a reduced fringing?

Have you already tried to compute the fields from some charge distribution?
If not, you should give it a try.

Michel

PS: my comments with "div E = 0" is just an other equivalent mathematical translation of the same thing

 

[tommyers]

Hi Michel,

I have finally managed to create a computer model in MATLAB which calculates the charge densities on either plate of a parallel plate capacitor. However my MATLAB 'mesh' only shows graphically the distribution over one plate (see BEMplate.jpeg).

The second jpeg (see BEMplatelines) shows what I hope to be my next step - modeling the field lines. My thinking is that now I now the charge at any (discrete) point on the plate then I could create a theoretical 'search' charge, say 1C that I could place at known vectors around the plate (x,y,z) from (x,y,z) on the plate. I now the two charges and their relative positions, so from this I could calculate the electric field at that point?

I have read many websites - and some of the theory is beginning to 'stick' better as I have been experimenting. I have 'The fundamentals of electromagnetism' by David Cheng - this is proving quite useful if not scary in some places!

What do you think to my above idea regarding electric field prediction? Do you think that from your last post that the fringing fields can be confined or reduced by other methods than making separation small? Using an Earth plane would be one idea, right?

Regards

Tom

 

[lalbatros]

Hi Tom

You should explain a bit more what you did, the method you used.
From your diagram the resolution of your calculations is not very high.
Have you used an analytical formula (more or less available here on PF from another post), or did you proceed fully numerically by BEM for example? If you wish, you can provide your source code: I could have a look and give you some advice.

For calculating the electric field, you don't need to place a charge or something like that. The electric field is independent from any probing charge, it depends only on the charges producing it. Here, the charges producing your field is the charge distribution on the plates. Calculating the field implies a sum (integration) of the effect of each charge elements distributed on the plate: this represents a substantial amount of calculations, but with today's computers this is quite feasible.

Note that you might improve a lot the efficiency of your modelling by taking the cylindrical symmetry of your problem. Because of the symmetry, the charge density depends only on the distance from the centre. Therefore you may benefit from working with "annular" element of charges: thins rings of charge. If you can find out an analytical formula for the field produc by such a ring you could boost your computational capacity.

Please give more detail on your calculation to facilitate a discussion.

Other topics to be discussed:

why do you want to reduce the fringing field ?
why do you think a ground plate would make things better ?
​

Michel

 

[tommyers]

Hi Michel,

I used a BEM method, as described in the document by Fred Erickson:

http://www.ttc-cmc.net/~fme/captance.html"

The parallel plate postscript is too large to 'attach' to this post!

I have basically followed this example through and converted the mathematics into MATLAB and then plotted the charges on a surf plot
(see attached MATLAB code (.m) file!)

The capacitors dimensions are very conceptual; plate area = 16m^2 and the separation is 10m! But I wanted to prove that I could follow the example and understand the thinking behind the document. - I was then hoping to use the same technique for my experimental capacitors, to compare the measured capacitances with the BEM capacitances with the capacitance calculated by the elementary capacitor equation.

I understand that the field is independant of the probing charge! I want some method of knowing how far the fields reach, how strong they are and what percentage they contribute to the capacitance, so that I would then know that at a certain distance from the plate edge although a fringing field may exist it could be ignored as it only contrbuted say ~2% to the capacitance.

I am open to any techniques providing they are flexible enough for different geometries and providing that enough information exists that I can use to get me up and running with the technique!

Fred Erickson has provided much support and conversation regarding his technique!

I am interested in looking into the suggested 'cylinderical' technique described. How you any links or further pointers to give?

Other topics to be discussed:

why do you want to reduce the fringing field ?
why do you think a ground plate would make things better ?

I don't want to reduce the fringing fields as such - I just would like for them to be accounted for in my calculation stage. It would also like to know what effect they have for each given geometry. It may even become the case that I would like to utilise them i.e. fringe field sensor.

why do you think a ground plate would make things better ?

Again, I don't think a ground plate would make things better in the parallel plate geometry - although I do if I were to implement a fringe field sensor geometry. As I would want the fringing fields to by unidirectional.

I look forward to your thoughts and comments.

Regards

Tom

 

[lalbatros]

Tom,

You should comment/explain your code a little bit more.

For example, I am not sure what (why) you call the "large potential equation". Concerning variables "potential_1_sig_1" and the like, explain you notation, are these the kij matrix from equation (40) in Eriksson? If your answer is "yes", then, could you explain your first relation:

potential_1_sig_1 = (k*((4*log(1 + sqrt(2))/h) + (1/(3*(sqrt(2)*h))) + (2/(3*h))))​

I would expect only one term, not three terms. What is your explanation?
Next, what is then the meaning of your next equation:

charges = [potential_1_sig_1 ... potential_22_sig_22]​

would that be the rhs of equation (39) in Eriksson? And the meaning of this one:

charge = charges\v​

is it the solution of equation (39) in Eriksson?

My other comments:

You did not describe the geometry you considered, square, circular, ... .
You did not give the coordinates of the surface elements. It could help you a lot to put that in variables.
You should make more use of vectors, for storing coordinates of surface elements, for automating the building of equation (39), ...
With more automation, you will be able to handle more elements and be more realistic.
Try to solve the problem with -say- 2*100 elements, once you finished this small one.
Then, try to understand a few questions of you own. My own main question would be: the role of the log terms (equ 40). What would be yours?
Calculate the potential and look at its behaviour very close to the plates and more away (equ 43).
Calculate similarly the electric fields (write your equation).
How would you define the effect of the fringing on the capacity and calculate it.

Enjoy,

Michel

 

[tommyers]

Michel,

You should comment/explain your code a little bit more.

Yes you are correct the commenting is scarse - I shall improve on this!

Concerning variables "potential_1_sig_1" and the like, explain you notation, are these the kij matrix from equation (40) in Eriksson? If your answer is "yes", then, could you explain your first relation:

potential_1_sig_1 = (k*((4*log(1 + sqrt(2))/h) + (1/(3*(sqrt(2)*h))) + (2/(3*h))))

Yes. This is based upon the kij matrix in equation(40) but more directly equation(45). I have separated each potential 1, 2, 6, 17, 18 and 22 and then further each charge associated with that potential 1, 2, 6, 17,18 and 22.

Thus, potential 1 and charge 1 has the notation 'potential_1_sig_1' and is equated to the terms made up from the kij matrix previously discussed.

I would expect only one term, not three terms. What is your explanation?

Why?

Next, what is then the meaning of your next equation:

charges = [potential_1_sig_1 ... potential_22_sig_22]

This is a method in MATLAB to create a matrix, there are six unknowns i.e sigma1, sigma2, sigma6, sigma17, sigma18 and sigma22 therefore there are six equations in the matrix potential1 through to poential22.

And the meaning of this one:

charge = charges\v

This again is a MATLAB operation. Basically it performs a simultaneous equation using the matrices charges and v (the potential vector).

My other comments:

You did not describe the geometry you considered, square, circular, ... .

This was the parallel plate arrangement as described in Ericksons document. It consists of two sqaure parallel plates 16m^2 by 10m separation.

You did not give the coordinates of the surface elements. It could help you a lot to put that in variables.

These are described in the document, is that what you meant?

You should make more use of vectors, for storing coordinates of surface elements, for automating the building of equation (39), ...

I like the sound of that!

With more automation, you will be able to handle more elements and be more realistic.
Try to solve the problem with -say- 2*100 elements, once you finished this small one

Yes this is my goal - I would like to be able to simple modify some standard code for different geometries. The more realistic the better. However I am only requiring a resolution to that of my capacitance meter i.e 0.1pF.

2*100 elements

Is the 200 elements? - If so why have you written in such an interesting way?

once you finished this small one

Don't you consider this finished?

Then, try to understand a few questions of you own. My own main question would be: the role of the log terms (equ 40). What would be yours?

I haven't thought that far in advance yet - although I presume that the log terms have something to do with the 'self' capacitance or the mutual capacitance. I am not 100% sure what these are when they are described by Erickson. Do you?

Calculate similarly the electric fields (write your equation).

Do you think this technique lends itself to field distribution instead of charge distribution?

How would you define the effect of the fringing on the capacity and calculate it.

I was presuming that the charge and the fringe were related? and once I knew the charge at a point I could figure out information about the fringing field, such as, the 'strenth', its reach and by knowing these two its overall significance on the capacitance?

Thank you for your time to this discussion.

Regards

Tom

 

[lalbatros]

Hi Tom

I did not realize that you were treating the sample problem from Eriksson (equ 45), sorry. This settles about half of the questions. Mainly this explains why "potential_1_sig_1" contains 3 terms instead of one: the symetries (equ 44) reduce the number of equations and unknowns, and several kij are grouped together in compacted system of equations.

For the MATLAB stuff, no problem, MATLAB first versions date back to the 80's. The syntax is not funny.

Concerning more general calculations, you will probably discover that it is easier than re-typing equs 45. However, the number of unknowns increase as the square of the spatial resolution. Without exploiting some symmetry of the problem you will easily reach the limits of your MATLAB version (I assume it is a student version).

On this small problem, you can still make a lot of work, specially for checking the possibilities, your understanding, and others. You should calculate the potential and make contour plots of the potential in some cross section of the capacity. You should also calculate the electric field. You should also think to draw the electric field lines (how would you do that?). You could also check that the charges you calculated do indeed produce constant potential on each plates. You could also perform the potential and fields calculations with a uniform charge distribution and make a comparison. Personally, I would also perform the same calculations when droping the logarithmic terms and see what happens and think about it (this actually means replacing elements by point charges). All this can also be done on more detailled calculations. When you think you are finished, tell me!

From equ (43) it is clear that the method enables you to calculate the potential, and therefore also the electric field.

There are many other possibilities for thinking and study. For example, you might think about the respective benefits of using the analytical formula given here on PF and using the BEM method as discussed now. I still don't know. Another example is the possibility to specialize the BEM for a circular symmetry.

Michel

 

[tommyers]

Michel,

Please could you elaborate on this for me:

Concerning more general calculations, you will probably discover that it is easier than re-typing equs 45. However, the number of unknowns increase as the square of the spatial resolution.

Surely the equations given (the terms) are purely suited to the geometry alone?

Could you explain the phrase:

the potential in some cross section of the capacity.

and...

You could also check that the charges you calculated do indeed produce constant potential on each plates. You could also perform the potential and fields calculations with a uniform charge distribution and make a comparison.

When considering equation (43) this could be used to field the electric field at some point in space by making sigma a charge on an element of the plate and making dij the distance from the centroid of that charge to the point in space?

using the analytical formula given here on PF

Could you give me some pointers to there loactions?

Regards

Tom

 

[lalbatros]

Hi Tom

Quote:
Concerning more general calculations, you will probably discover that it is easier than re-typing equs 45. However, the number of unknowns increase as the square of the spatial resolution.

Compare equations 40 and 45 ! Using matrices in MATLAB together with equ 40 will produce a general programme easier than retyping the whole bunch of equations 45 ! In addition, we cannot be totally sure that no mistake popped up in 45 ! While starting directly from 40 should give you at least the same result (but more equation if symmetry is not exploited) without the headache.

Quote:
the potential in some cross section of the capacity.

You could map the potential in any plane, specially one perpendicular to the plates. A symmetry plane would be interresting.

Quote:
You could also check that the charges you calculated do indeed produce constant potential on each plates. You could also perform the potential and fields calculations with a uniform charge distribution and make a comparison.

Usually all models should be checked, at least for mistakes but also fo precision, limitations, ... In the case here many things should be checked.

Calculating the potential from 43 should give the right potential on the plates (note: comparing 43 and 40 shows that some care must be taken when on the plates). This will check the solution of the linear system (the "charges\v" line) as well as the expression copied from 45.

Comparing the solution with a uniform charge density is a little bit of physics. You will measure the importance/impact of the inhomogeneity of the charge distribution.

Additionally, it could be interresting to analyse your calculations as a function of the distance d. You may see that when d becomes small, the charge density become more uniform.

Quote:
using the analytical formula given here on PF

see https://www.physicsforums.com/showthread.php?t=125483, specially page 2.


Michel