This page walks you through designing your own DIY/homemade coil. I've done it for making coils for crystal radios and Tesla coils but it works for any coil that's cylindrical in shape. This is also useful is you're going to be using your coil in an LC tank resonant circuit.
An inductance calculator is provided below for making it easy.
Inductance is often what you are trying to achieve when designing a coil i.e. you know the inductance you want and now you have to design a coil that will have that inductance.
Coils have a property called inductance. What is inductance? When electrical current changes as it flows through the wire of a coil, it produces a changing magnetic field that induces (produces) a voltage, or emf (electromotive force), in the wire that opposes the current. That is called induction and inductance is a value quantifying the ability of the coil to induce that voltage. The symbol for inductance is the Henry and the unit is H. Here we're actually talking about the coil inducing the voltage in itself, which is self-induction, but we'll just say induction.
One formula for inductance is as follows:
* The inductance calculator below also takes inches, as well as centimeters and millimeters, and does the conversion to meters for you.
As said above, the μr is the relative magnetic permeability for whatever you're using for the coil's core, the cylinder that you're winding the wire on. That's the greek letter mu, μ, though often the letter u is used for convenience as in ur. If it's a hollow cardboard or plastic tube then the cardboard or plastic counts as air and you can use 1. Materials like iron and ferrite have higher relative permeabilities in the hundreds and thousands. For an iron core, a rough number to use is 100 though it really varies depending on the alloy. The same goes for ferrite which can have value anywhere from 20 to 5000 but if you're at a loss as to what to use then 1000 is a rough compromise. Since it's multiplied by the rest of the formula, that means using those materials will give a higher inductance value. Cores for crystal radio coils are sometimes plastic or cardboard, and therefore are air core coils, and sometimes they're ferrite core. The cores for Tesla coil secondary coils are usually plastic, and smaller ones may be cardboard, and therefore count as air core coils.
And if you're not familiar with the notation 1.26x10-6, it's just another way of writing 0.00000126.
Here's an inductance calculator that uses the above formula. The diagram above can be used as a guide for some of the parameters.
Video - How to Design a Coil for Specific Inductance
In this video I explain the formula for induction in detail, as well as give an example and talk about other factors like coil capacitance, frequency and coupling.