Resistances of 3 Dimensional Objects

[wagnerc]

Greetings. I have a very interesting problem that I don't even know where to begin to look for info. Or even what to type into Google. Good thing I found this forum. Very informative discussions. What I'm looking for is information on calculating the resistance through large objects. It's tricky because there's no regular current path like in the trivial case of a wire. For example, if you had a 3 inch square cube of some uniform substance, all of whose properties were known, figure out the resistance between any two points on that cube. Obviously the closer the test leads are the lower the resistance and vice versa. It reminds me of the force trajectories in stress analysis of metals. I have thought about simulating the cube as a very very large array of resistors in ultra series/parallel combo and writing a program to compute it but there's got to be some really cool way to derive it with calculus or differential equations. Anything appreciated, thanks.

Chris

 

[itsmoked]

You are talking just about DC right? Otherwise it gets even worse.

I think you could google for "bulk resistance"

 

[btrueblood]

You also need to consider contact resistance, both for your test lead connections and if you have any multi-body objects to analyse.

 

[VEBill]

"Obviously the closer the test leads are the lower the resistance..."

Not sure about that. In 2D, sheet resistance is measured in ohms per square (distance between the test points doesn't matter). I don't know off-hand about 3D bulk resistance, but I wouldn't say it is obvious. Google is your friend.

Contact resistance only matters if the material under test is relatively low resistance. Four wire measurements are one option for dealing with that if required.

 

[nbucska]

The simplest is to model it with a liquid filled tank
and use AC !!!

<nbucska@pcperipherals DOT com> subj: eng-tips
read FAQ240-1032

 

[IRstuff]

Partially true. Ohms/square is derived from the 4 point probe measurement that takes into account the distance between probes. The spreading resistance affects the total resistance, so the probe separation is used to normalize the measured resistance to the sheet resistance.

For a 3-D object, there would be a similar treatment, but would require you to do the transformation for the 3-D spreading resistance. I can't remember the original source, perhaps Van der Pauw, himself? The original article describes the mathematical approach for determining the mathematical treatment of 4-point probe measurements, where the probes are arbitrarily positioned. I haven't even looked at my hardcopy in probably 10 yrs, so my recollection is somewhat hazy.

TTFN

 

[fsmyth]

Narrow your search to PCB traces/conductors.
I vaguely remember reading an article on same
that delved fairly deep into the subject, and
the treatment of 3D objects was one of the
subjects. Sorry that I cannot recall more.
<als>

 

[VEBill]

Wandering slightly off-thread: The units of 'Ohms per square' isn't due to any particular measurement technique so long as you are measuring the simple resistance across an actual square shaped area. 4-point is just one way to accomplish that. I can't explain 'Ohms per Square' any better than this:

http://www.esdjournal.com/techpapr/ohmmtr/ohm.htm

Back to original question, see this Googled-nugget:

http://www.jandel.co.uk/faq.htm

 

[wagnerc]

I did manage to find one page dealing with the subject on a site dealing with fractal arrays. They even showed a formula for the relative resistances of 1, 2, and 3 dimensional objects. 1D, resistance doubles with length. 2D square, resistance independent of square size. 3D cube, resistance halves with length. With the square it seems that the increase in "series resistance" is exactly canceled out by "parallel conductance". With a cube the parallel is increasing more than the series so the resistance goes down by half. Not too useful for figuring out an answer to my problem but it does serve to codify a theory of mine. That the primary resistance is that of the straight line between the leads and that this is reduced by the parallel conduction going on adjacent to the direct path, reducing to an infinitesimal value the farther out u go. I believe that these paths could in fact be solutions to one differential equation describing the whole thing. The shape is actually rather similar to a magnetic field. The more obscure the subject, the worse Google does. I've tried all kinds of search phrases and nothing really good has come up. Anybody know any professors they could hit up for search terms or direct info?

 

[VEBill]

The Google-nugget previously linked provides a possible way forward (even for analysis):

"...if one already knows the sheet resistance, bulk resistivity can be calculated by multiplying the sheet resistance in Ohms-per-square by the thickness of the material in centimeters."

 

[nbucska]

Is this a practical or theoretical problem? If later,
how much accuracy do you need?

I suggest the following approach: make a math. model,
integrate it and check it with the liquid tank.




<nbucska@pcperipherals DOT com> subj: eng-tips
read FAQ240-1032

 

[IRstuff]

Ohms/square only works for uniform layers of material. That's why metal and diffusions are specified in ohms/square. Bulk materials are always specified in ohm-cm.

What is referred to in the reverse calculation is the bulk resistivity of a LAYER, not a bulk material.

TTFN

 

[wagnerc]

Just knowing the bulk resistivity value itself won't help me. I need to figure out a way to compute the total resistance when given the resistivity, lead size, and lead location alone. The size because the area of contact will affect the total value. This is in fact a practical problem. I'll eventually need to figure out resistance through irregular objects defined by continuous curves. Can anybody recommend some good simulation software? To model it it would have to be able to resolve an array of thousands of resistors.

 

[VEBill]

I wonder if there might be some off-the-shelf software for a parallel (similar) 3-D linear problem like heat flow or ???.

 

[btrueblood]

It's been awhile, but a linear FEA model that handles heat diffusion should also model electrical diffusion (flow of current thru the 3-d body); the equation for heat flow q = kA/L (T1-T2) is analagous to the electrical resistance equation i = A/(L*rho) *(V1-V2). You need to play with the units you put into the modeller to get the correct outputs, but a simple model of a wire with voltages applied at each end will be useful for getting the right numbers. Again, modelling the contact resistance for your connection details will be the toughest part, and will be the greatest source of errors when you go to test, IMO.

 

[VEBill]

...and it will be so much fun to present the results derived from such an unexpected SW tool.

 

[wagnerc]

Actually any good electric sim should do it - provided it can handle the quantity. Any 3 dimensional structure can be represented 2 dimensionally if necessary. In fact any n dimensionally array can be represented unidimensionally, but I digress. I was thinking Spice for the sim but can it handle 1,000 - 10,000 elements? And can it take text input from a file? I would write a script to generate that. I've also heard references to matrix methods from solving arrays of resistors. Anybody know what that is?

 

[IRstuff]

You can write your own in your programming language of choice.

TTFN

 

[abcd3286]

I am curious as to why you want to do this.
What is the practical real world application?

My first thought when I read this was calculating the resistance of bus work.

Dan Bentler

 

[wagnerc]

Heh, I was wondering when somebody was going to ask that. The truth is, it's sort of a secret. It has to do with the applications of shaped resistive bodies. If I say too much somebody'll surely rush down to the patent office and screw me. :-D

chris