Multilayer air core coil optimization
First of all I'd like to mention that I'm not a professional in the electromagnetic field, so please excuse me if I make major mistakes.
A while ago I was confronted with the problem of designing multilayer air core coils for speaker crossovers. The inductance for MACCs (multilayer air etc...) is computed with Wheeler's formula. By looking at the formula one can notice that a specific inductance can be obtained with different coil dimensions (internal/external diameter and height), and different wire lenghts and resistances implicitly. So, the question of optimum coil parameters (dimensions), in the sense of minimizing wire length and resistance arose. So, I tried computing these optimum dimension analitically, just to realize that this leads to a 12 order polinomyal equation (there probably is an easier way of doing this but I missed it). Than I wrote a Matlab script to compute these params numerically, and I got to some simple formulas. I noticed that the ratios of the dimensions remain constant, in other words the "shape" stays the same, only the size changes with required inductance and wire diameter. The approximative results I obtained are:
- height is approx 1/4 the external diameter
- internal diameter is approx a half the ext diameter
So, now my questions.
1. I don't think I'm the first to do that, and I'm really curious what is the correct way to do this analitically.
2. Is there any downside in building an "optimized" MACC? I mean... I don't know how the Wheeler formula is obtained, but the formulas they taught us in school always had volume integrals in them and the Wheeler formula is a simple fraction of polynomials, so I guess it's the approximation of an integral. And I also guess that the real inductance of a 1 meter wide coil with an 1 inch height differs from the Wheeler approximation by much, if you get my point.
Sorry if I was such a bore, and thank you in advance.
It's funny that noone even bothered to tell me that what I calculated is called "the Brooks inductor". Usenet rules in this respect btw
The reason no one was familiar with your request is that as electronics gets more complex, component design becomes a specialized function and the practitioners are very few and work for component manufacturers. Such people would not be members of this site which deals with circuit and system design.
Another dimension of complexity of inductor design is parasitic capacitance minimization. In this case the style of the winding and the space between turns becomes important. For maximum working frequencies and maximum Q, such coils are single layer and have diameters about twice the length.
Since these coils are used for speaker crossovers, at audio frequencies, probably parasitic capacitance is not an issue. I really can't imagine how big would a single layer 3mH air-core coil be!