Wire through toroid capacitance

[bubulescu]

Hello,

I'm in a small dilemma. I'm making a model of a Rogowski current transformer in LTspice and I'm done with it, except I don't know how to model the capacitance between the wire running through the toroid. I have the inner resistance, capacitance, almost all needed for the secodary inductance, but I would like to model this one, too. Now, I know it is virtually negligeable, but for inner radii small enough, the effect is present, albeit in small numbers.
I've attached an image which is suppopsed to replace a thousand words, English not being my native language. You're looking at a simplified section view of the left half: in the left side there's the circular section of the bobbin and to the right the conductor. What I want is to find out what is the medium distance (marked with x) between the wire running through the center of the bobbin and the inner surface of the bobbin. Having that, I can calculate, or approximate close enough, the parasitic capacitance, as being a conductor in a cylinder. The formula, in this case, would be:

 2×π×ε0×l/ln(2R/Ø)

where R is the outer radius, Ø the conductor's diameter and l the length of the conductor. Of course, the condition would be for the conductor to run through the center of the toroid, but that can be ignored since it's a simulation.

So, mathematics not being my strong point, I am now asking for your help.

 

[FvM]

Why don't you use paper, pencil and a ruler? The capacitance approximation is incorrect anyway, because the real geometry
is different. If a mathematical problem, it would be worth to be solved exactly, but it isn't.

A different question would be to determine the capacitance exactly. That's possible in principle, rotation symmetrical problems
are usually handled by conformal mapping. But today, most engineers prefer a numerical tool. Quickfield is quite good for similar
2.5D problems, full featured EM tools as ADS can of course.

If you have a problem with capacitive coupling, a differential sense amplifier respectively integrator would cancel it. In most
Rogowski designs, the internal "coil" capacitance is a more severe problem, because it sets a limit to the bandwidth.

Also a screening cylinder wouldn't affect the Rogoswski sensor operation.

 

[bubulescu]

Why don't you use paper, pencil and a ruler? The capacitance approximation is incorrect anyway, because the real geometry is different. If a mathematical problem, it would be worth to be solved exactly, but it isn't.

I used those and I came up to this approximation: (R-r)²(8-π)/32 but, as I said, I do not trust my math. However, the formula for the capacitance is correct and can be approximated to that, even if the real geometry means radial turns which would decrease the capacitance. But if I were to calculate for a whole surface, the effect would be that of the worse possible case - the resulting capacitance would be higher.

A different question would be to determine the capacitance exactly. That's possible in principle, rotation symmetrical problems are usually handled by conformal mapping.

I could calculate relative to one half of a turn, then multiply by N, and the result would probably be close to reality, but the formula and the amount of computing for such a negligeable effect would not be justified.
Suppose there is a square section (which would make a good surrounding cylinder for the wire running through) with R=50mm, r=25mm and h=50mm , the parasitic capacitance would be ~1pF (1.066, calculated, approx.), compared to the ~85pF of the second inductance. For such a geometry and such a small capacitance, the effect is negligeable, indeed, but the whole purpose of the simulation is to give an idea of the real-life situation because there are so many unknown, therefore an exact calculation is, always, impossible.

But today, most engineers prefer a numerical tool. Quickfield is quite good for similar 2.5D problems, full featured EM tools as ADS can of ourse.

This would complicate things even more. I can only imagine how proud the parasitic capacitance would feel knowing that someone is going through that much trouble just for it

If you have a problem with capacitive coupling, a differential sense amplifier respectively integrator would cancel it. In most Rogowski designs, the internal "coil" capacitance is a more severe problem, because it sets a limit to the bandwidth.

I know, but here is no problem with the coupling, it's only an extra parameter that I would have liked to add so as to be closer to reality, nothing more. The results until now prove that I may as well omit it. Also, an integrator other than that of the "restoring" the signal's phase is out of the question, since afterwards I'm forced to take precautions of the phase, too, which is always a pain in measurements.

Also a screening cylinder wouldn't affect the Rogoswski sensor operation.

True, but that would add a lot more to the self-capacitance than it would to the coupling - bigger surface, not to mention the difficulty of calculating that, too. As a practical measure, sure, it is welcome for high curents.

Well, if it's one (obvious) thing that's to learn from here is that one must be careful when hunting details - they might not be worth the trouble. Thank you for the answer(s).