I spent the entire day and night yesterday and most of today working through another set of math based upon research by two of the three authors from the book Axial flux permanent magnet brushless machines
(Ref. 1). Part of the book is published online as sample content; preview of what’s inside sort of thing. The segment of interest though is not, however I found an independent publication of work on IEEE which loosely covers the same material as the book, though in more depth (and I am glad to have found it).Analysis and Performance of Axial Flux Permanent-Magnet Machine with Air-Cored nonoverlapping Concentrated Stator Windings
The math here validates that my model is good, and according to the authors’ formulations I should expect a little better than what FEMM is reporting; certainly that is welcomed news. However there are some issues that are not well documented – leading to assumptions that mount up in very short order. As I said, I spent day and night and another day on this, but was completely stuck on one section and interpretation. Allow me to explain:
If you have the book (Ref. 1), look at the Numerical Example 5.1
: These are nearly identical except for a couple of small changes to the pole count and spin. The IEEE covers much of the same formulations albeit not identically. Let’s take 5.1
m1 = number of stator phases = 3
p = number of pole pairs = 3
s1 = number of single-layer coil sides (equivalent to number of slots) = 54
wc = coil pitch measured in coil sides = 7
The first and biggest obstacle was how to interpret coil sides: In the book, wc = coil pitch measured in coil sides
, and in the IEEE article (Ref. 2) it’s w = Width of coil side in meters
that is later determined through angular measurement and average magnet radius. The problem is that the value given in the Book example does not compute and I cannot find a decent explanation for how they have derived the value. Review of Numerical Example 5.1
54 coils / 3 phases = 18. For ironless air-cored stators the book likes to count coil-pairs as a complete coil, therefore 18 / 2 = 9 coil pairs in a coil phase group. How do the authors get wc = 7?Numerical Example 6.1
is the same as 5.1
p = 12
s1 = 72
wc = 3
ksat = 1 because this design is ironless.
Again, I cannot fathom how wc is calculated. In the hunt for the definition of coil pitch I came upon this article:COIL PITCH IN AN AC ARMATURE WINDING FULL PITCH WINDING
Helpful, but doesn’t agree with how wc is calculated. I found yet another paper from the same South African university as the Ref. 1 Books’ authors, and this individual was far less cryptic: Design, Comparison and Experimental Evaluation of Non-Overlap Winding Radial Flux Permanent Magnet Hub Drives for Electric Vehicles
Ns = number of stator slots
Nl = 1 for single-layer and = 2 for double-layer windings.
Q = number of stator coils given by NlNs/2
Given that pp = pole-pairs, and gcd = greatest common divisor,
Ms = number of machine sections = gcd(pp, Q)
Ws = number of winding sections = gcd(2pp, Q)
u = number of coils distributed in a coil phase group = Q/mWs where m = 3 phases
S = number of stator slots per machine section = Ns/Ms
I went through all of this and kept going up to Page 13
of Ref. 4 where he’s talking about slot fills and it dawns on me what the coil width is all about: There is no calculation for wc; it’s a ratio.
The problem is that the ratio for the original problem does not compute in my model and because of that – the eddy current calculations are wickedly wrong which later impacts the model efficiency calcs. For my model, wc = 4, 7, and 10 had positive effects though each had the exact same Pout rating, whereas any other number below 10 adversely affected power. No single number agreed with the original calculations. At the time of this writing I was at a loss to understand why (although now I have a good grasp).
So far I am pleased with the results; arduous and detailed – yes, and there is still more work left to do before cutting steel as they say. I still have to consider the inertia, and that may weigh against the nice efficiency that’s developing.
One last note: After doing the math
on three similar models this weekend, I now have a better understanding about eddy currents
. I get it; this was a most useful exercise. On paper, the new version (I suppose we could call it Plan-F
for “fun”) has a kT >1.8 Nm/A, kV of 14 rpm/V, and Efficiency of 94% @ the rated speed. It’s a little heater though; would be great to tap into that during the winter and pipe it up into my all-weather suit.
More twiddling is required. Maybe tonight I'll be able to sleep