Air coil cores with chart to determine inductance

Use case

Air coils give the best possible Q factor in L-C filters or oscillators, as they are used in radio technology by commercial manufacturers and amateur radio hobbyists. For private projects they are extra convenient, as they can be homemade and wound for the needed inductance. So you don't need to purchase a complete assortment, to be prepared for that Sunday afternoon building project. 

This publication describes a way to very predictably produce the coil for the inductance that you need. 

Wheeler Formula and Air‑Core Coil Theory

Air‑core inductors are widely used in RF applications because they avoid magnetic core losses and behave predictably at high frequencies. Their inductance depends mainly on the coil’s geometry: diameter, length, and number of turns. A well‑established approximation for single‑layer air coils is the Wheeler formula, which expresses the inductance 𝐿 in microhenries as:

where 𝑟 is the coil radius, 𝑙 the coil length, and 𝑁 the number of turns (all dimensions in inches). The formula shows that inductance increases with the square of the turns, but also that a longer coil reduces magnetic coupling between turns. As a result, when the pitch is fixed and the coil becomes longer with each added turn, the growth of inductance becomes almost linear rather than quadratic. This makes the Wheeler formula an excellent predictor for practical RF air coils and allows accurate design of custom inductors based purely on geometry.

Theory vs. results achieved in reality

The cool thing is that the above stated formula predicts your real-live results very accurately, if you manage to achieve a very precise mechanical realisation of the coil. This is where these little printed cores come in. The cores have tiny holes that allow to have any number of full or half turns and fixate the wires ends nicely and precisely. They also have a winding groove, just like a thread, to very accurately guide the wire around the core and ensure very evenly spaced windings.

Have a look a the chart below. It shows the theoretical inductance value from the above formula, and my achieved results. I have done my measurements by putting the coils into a series L-C notch filter with 4 different known capacitors, and measuring the resonance frequency with my NanoVNA-F V2. From that I calculated the back to the inductance value and took the average of the 4 values (the spread was small, somewhere around +-5%, tracing back to the tolerance of the used capacitors). 

As you can see, my results are basically perfect hits. That means that also you can use these cores and be sure to achieve the calculated inductance, which is especially important if you have no means of measuring the inductance yourself. 

Printing

These cores are very simple and very fast to print. Just print them in the uploaded orientation. I recommend to print at least 4 at a time, to give each layer a bit of time to cool down. I have used PETG as material, but I am fairly sure PLA is ok as well (you need to be fast when soldering the coil to your application, as PLA gets soft very quickly when it gets warm, with PETG soldering was not problem, just be a bit quick). I recommend 3 perimeters, to make sure it is all perimeters (all concentric). I was a bit careful and have used brims to make sure the cores don't tip over while printing, but with good bed adhesion that is likely not even needed. Obviously, supports are not needed.

Assembly

I have used copper wire with 0.6mm diameter that was silver coated (i.e. not isolated, as that is not needed on these cores). You can use any kind of wire that you manage to stick through the tiny holes. Stick the wire into one hole and out on the opposite site. Then wind the wire very closely, following the groove. You can end the coil after each half turn, by sticking the wire through the core once more. My measurements where all done with coils wound like this. 

Example: Coil with 10 turns: Stick the wire through the core, then start winding and count exactly 10 full turns, then stick the wire through the core again. Both ends of the wire will stick out on the same side of the core. Again, make sure that the wire is very tightly following the groove in the core. 

Please let me know how these coils work for you

I would be delighted to learn about your projects and how these coils have worked for you. If you like this project, please leave me a like and post you make.

Happy printing!