Electronic – Approximate Capacitance of a Multilayer Solenoid Coil

capacitancecoilparasitic-capacitance

I'm interested in an expression for the capacitance or resonant frequency of a Helmholtz like coil (the windings are parallel to the plane of the coil) of circular or square cross section and of the order 1 meter in diameter with between around 10 to 100 turns.

I can find numerous approximations for the inductance of such coils, but I can't seem to find any approximations of capacitance.

I suppose this will vary wildly based on the winding technique and frequency, but I'd be interested in any expressions for this.

This answer contains an expression for the capacitance of a single layer solenoid which is almost what I need.

Best Answer

I used to have trouble estimating the self capacitance of coils, until I came across a simple change of perspective (don't know where I read it) that basically says 'apply a voltage across the coil, and compute the stored energy due to capacitance'. It doesn't sound like much of a simplification. However, it does emphasise that the bulk of the energy is stored where there's a large voltage between turns.

This means that if you have a multi-layer coil, then a major simplifying assumption you can make is that the adjacent turn capacitance is negligible compared to the layer to layer capacitance. If you have (let's say) a 10 turn layer, then the inter-layer space stores 100x the energy of the inter-turn space, volume for volume, and so dominates the calculation.

The estimation runs as follows. Treat each layer as a conductive sheet. For each adjacent pair of sheets, compute the mean squared voltage between them (see further down) and then use the parallel plate capacitance formula to estimate their contribution to the stored energy. Obviously you have to estimate an effective spacing for the layers, being made of wires, but you do arrive in the ball-park by this method, dealing with only a handful of layers, rather than 1000 turns.

This leads on directly to several methods to minimise self capacitance.

  • If you have spare space in the bobbin, should you space the turns on each layer, or pack them tight and space the layers with more inter-layer tape? Obviously the latter.

  • Compare back-and-forth winding with uni-directional winding. The former is easy to do. The latter requires you run a return wire back to the start between layers, and insulate it above and below, is it worth it?

Consider two 5-turn layers, the first wound back and forth, the second uni-directional, ascii-art shows the voltage on each conductor on a section through the coil

back and forth            uni-directional

0  1  2  3  4             0  1  2  3  4
9  8  7  6  5             5  6  7  8  9

The energy stored in the first configuration is proportional to \$9^2+7^2+5^2+3^2+1^2 = 165\$, the energy stored in the second to \$5 \times 5^2 = 125\$. These sums give you a hint as to how to compute the mean square voltage between layers. So depending on how easy it is to wind unidirectional, it may be worth it, and it always is if you need to scrape the last bit of SRF from a coil.

  • If you have opened an old radio, you may have seen inductors wound with pancake coils, and in fact many SMPS transformers are wound on sectional formers.

Using the same ascii-art presentation for these, their windings would look like

 4      9
3 2    8 7
0 1    5 6

Without doing the calculations, it's clear that the inter-layer voltages are less than for the wide layer cases above, and so the capacitance will be less.

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