# Approximate Capacitance of a Multilayer Solenoid Coil

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.

• give some dimensions and SRF value and proximity to ground plane. May 15, 2018 at 2:22
• About 1 meter in diameter with 10 to 100 turns. If I had the SRF I would already have the answer. I'm interested in the parallel capacitance of this coil so I don't think the ground plane distance matters? In any case I don't have one yet. May 15, 2018 at 18:53
• So the wavelength is ? May 15, 2018 at 20:38
• You cannot measure capacitance if there is no ground plane , therefore it is transmission line reflection and magnet wire fill factor dielectric constant . May 15, 2018 at 20:45
• At sufficiently high frequencies the coil's adjacent turns will be at different potentials causing energy to be stored in the coil capacitively. I don't think the ground plane has a significant impact on this effect. May 15, 2018 at 23:37

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.

• I once tried for a Rogowski coil, and ended up at a turn level, where the turn has a (half) circular section against the geometry it's wounded on. The section is basically sqrt(1-x^2), the integral pi/2, the rectangular section holding it is 2, so the ratio is (pi/2)/2=pi/4 as an average length. Using this as a starting point proved to be close enough for determining the self capacitance. If this holds up, then, between two turns on different layers (ideal case), the mean length is (pi/4)*2. Of course, gross assumptions, but, it was a starting point. May 15, 2018 at 6:15
• This is a good idea. I think can turn this method into continuous functions for various winding patterns and cross sections. May 15, 2018 at 19:03

A Helical air core inductor can resonate at it's shorted 1/4 wave or open 1/2 wave length. Depending on gap, pitch or length of wire l/d ratio and many other factors, you get the idea that computing self-capacitance is a bit more than Sacred Geometry . There is no one simple formula.

H. F. Resistance and Self-Capacitance of Single-Layer Solenoids", R G Medhurst (GEC Research Labs.). Wireless Engineer, Feb 1947 p35-43, Mar 1947 p80-92.

This is a big read but you won't get accurate results without a lot of researchreported here, with abstract below.

# Inductor self-resonance and self-capacitance

The self-resonance and self-capacitance of solenoid coils By David W Knight

Self-resonance and self-capacitance of solenoids . DOI: 10.13140/RG.2.1.1472.0887

• Open Document spreadsheets and coil data F61-32T.ods , Medhurst.ods , CL_theor_test.ods , 18T_scat_Howe.ods , CL_axial-field.ods . Helical_vf.ods . Maxi Spring Air Core Inductors. Coilcraft Document 185-1, 2004 (accessed 5th Jan. 2016)

## velocity factor comparison

http://g3ynh.info/zdocs/refs/Medhurst/Medhurst1947.html

## Abstract:

The data on which Medhurst's semi-empirical self-capacitance formula is based are re-analysed in a way that takes the permittivity of the coil-former into account. The updated formula is compared with theories attributing self-capacitance to the capacitance between adjacent turns, and also with transmission-line theories. The inter-turn capacitance approach is found to have no predictive power. Transmission-line behaviour is corroborated by measurements using an induction loop and a receiving antenna, and by visualising the electric field using a gas discharge tube. In-circuit solenoid self-capacitance determinations show long-coil asymptotic behaviour corresponding to a wave propagating along the helical conductor with a phase-velocity governed by the local refractive index (i.e., v = c if the medium is air). This is consistent with measurements of transformer phase error vs. frequency, which indicate a constant time delay. These observations are at odds with the fact that a long solenoid in free space will exhibit helical propagation with a frequency-dependent phase velocity > c. The implication is that unmodified helical-waveguide theories are not appropriate for the prediction of self-capacitance, but they remain applicable in principle to opencircuit systems, such as Tesla coils, helical resonators and loaded vertical antennas, despite poor agreement with actual measurements. A semi-empirical method is given for predicting the first self-resonance frequencies of free coils by treating the coil as a helical transmission-line terminated by its own axial-field and fringe-field capacitances.