## Doing the Math

Electric Motors and Controllers

### Re: Doing the Math

I've tried your calculator and it worked fine!

I think it would be wise to start calculating with the 2.53 m as only the part of the wire that is peripendicular to the magnetic field.

/GÃƒÂ¶ran

Goethe
10 W

Posts: 75
Joined: Tue Dec 08, 2009 7:33 pm
Location: In Sweden with polar bears and Santa as neighbours

### Re: Doing the Math

Gotcha. Let me state that I have set up a support page in the Technical Reference section the Archimedes-Winding Calculator.

Resuming where I left off, we have the following:
A 24-inch/0,61 m diameter wheel spinning at 30 mph/48,3 km/h providing 2 hp, and a single circuit carrying 44A at 34.4V along a single 11AWG conductor that is 2.53 m in length.

Using the nifty calculator, we can deduce the following, given:

w = diameter of the wire = 2.30378 mm
ID = Inside Diameter = 15 mm
Ag = AirGap = 0
l = Length = 2.53 m
YieldsÃ¢â‚¬Â¦

OD = Outside Diameter, mm = 87.8634418888889
n = Number of Turns = 15.3138888888889

Now we can go back and attempt to correctly calculate Inductance (L) using inches with the formula given:

L (ÃŽÂ¼H) = A^2 * n^2 / (8A + 11 w), or more simply
L (ÃŽÂ¼H) = (nA)^2 / (8A + 11w)
Solving for A:

Note Ã¢â‚¬â€œ A is in inches, therefore
OD = 87.86/25.4 = 3.459
ID = 15/25.4 = 0.591
w = 2.30378 = 0.0907
A = [((OD Ã¢â‚¬â€œ ID)/2) + ID] /2 =>
[((3.459 Ã¢â‚¬â€œ 0.591)/2) + 0.591] /2 =>
A = 1.0125
Solve for L:

L = ((1.0125)^2 * (15.3144)^2) / ((8 * 1.0125) + (11 * 0.0907)) =>
L = 240.43 / 9.0977 = 26.43 ÃŽÂ¼H
Note that this is different by a factor of 2.5 than what is given in the calculator link below:
And is off by even more with this one:
However it is nearly the exact same with this calculator:
Solving for L in meters, the formula given on Wikipedia is:

L = (A^2 * n^2) / ((2A + 2.8d) * 10^5 =>
L = (An)^2 / ((2A + 2.8d) * 10^5
Where d = depth of coil; (OD Ã¢â‚¬â€œ ID) / 2

OD = 87.86/1000 = 0.0879
ID = 15 = 0.0150
d = (0.0879 Ã¢â‚¬â€œ 0.0150) /2 = 0.03645
A = [((OD Ã¢â‚¬â€œ ID)/2) + ID] /2 =>
A = (0.03645 + 0.0150) /2 = 0.025725
Solver for L:

L = (0.025725^2 * 15.3139^2) /[(2*0.025725) + (2.8*0.03645)]*10^5 =>
L = 0.155197 / 15351 = ridiculously wrong.
Conclusions:
I am at a bit of a loss for a decent formula for open air coil Inductance. Practical sources say to measure it in situ. Can we presume that we can affect the overall inductance with an inline coil to match our controller?

One step back before we take two steps forward.
~KF
* My 2WD Garden Wall
* Current ride: 2WD Disc EBikeKit (9C 2806-equivalent) / Dual Lyen 12FET / 15S6P LiPo when commuting.
* Going to California: 2011: Trip completed
* Club Member: 40-mph & 101. 10k-Club: 10,478 miles-to-date, 4,988 as 2WD.

It is by caffeine alone I set my mind in motion.
It is by the beans of Java that thoughts acquire speed.
The hands acquire shakes, the shakes become a warning.
It is by caffeine alone I set my mind in motion.

Kingfish
1.21 GW

Posts: 3867
Joined: Wed Feb 03, 2010 11:23 am
Location: Redmond, WA, USA, Earth, Sol, Orion–Cygnus Arm, Milky Way. Age: > yesterday < tomorrow

### Re: Doing the Math

Bypassing the Inductance (L) conundrum for a moment, I would like to focus on how to reduce the current through the conductor with the goal to reduce potential ohmic heating.

Our single circuit presently pulls 44 A. The most straight-forward method I can think of to reduce current is to add more magnetic poles to our design. However in doing so we must also up the A/C signal frequency proportionally to compensate.

I havenÃ¢â‚¬â„¢t been able to locate the precise formula, however we can assume that

ÃŽâ€ Magnet Pole pairs (p) = ÃŽâ€ Signal Frequency (f)
An example of that is displayed in this wiring diagram:

Therefore, if our spinning wheel is already rotating at 7 Hz and pulling 44 A, determine the changes when we apply a 7X reduction of current. Given:

Ii = 44 A
pi = 2
fi = 7 Hz =>
If = 44/7 = 6.3 A
pf = 2*7 = 14
ff = 7*7 = 49 Hz Ã¢â€°Ë† 50 Hz (European A/C frequency)
Realistically, the frequency can go up quite a bit higher before we would see appreciable losses; however it is doubtful this system will ever see a signal beyond 400 Hz even at WOT.

Another way to reduce the current is to divide the single circuit into 3-phases. Converting the current value from a single-phase to a 3-phase is given as I1p = I3p / Ã¢Ë†Å¡3 Ã¢â€°Ë† I/1.732

To convert our original 44 A single circuit to 3-phase, each individual phase would run at 25.4 A (a strange coincidence). If we took our newly-minted 7-pole paired motor and split it into 3-phase, the 6.3 A value would drop again to 3.6 A per phase.

These changes allow for a robust copper fill Ã¢â‚¬â€œ and we havenÃ¢â‚¬â„¢t yet optimized the coil windings and distribution.

Am I good so far?
~KF
* My 2WD Garden Wall
* Current ride: 2WD Disc EBikeKit (9C 2806-equivalent) / Dual Lyen 12FET / 15S6P LiPo when commuting.
* Going to California: 2011: Trip completed
* Club Member: 40-mph & 101. 10k-Club: 10,478 miles-to-date, 4,988 as 2WD.

It is by caffeine alone I set my mind in motion.
It is by the beans of Java that thoughts acquire speed.
The hands acquire shakes, the shakes become a warning.
It is by caffeine alone I set my mind in motion.

Kingfish
1.21 GW

Posts: 3867
Joined: Wed Feb 03, 2010 11:23 am
Location: Redmond, WA, USA, Earth, Sol, Orion–Cygnus Arm, Milky Way. Age: > yesterday < tomorrow

### Re: Doing the Math

To increase the efficiency in a motor without total redesign I would start to look at the magnets and airgap.

I have been going through my papers and some stuff I've found on the web and I think we have to restart this whole thing from another angle.
The magnetic flux is built up from many components and to get decent figures these components canÂ´t be left out. For example is the permeability constant about 7000 times higher for an electric steel core compared to the air like in the coil we calculating with now.
Magnet data, Air gap geometry, teeth geometry, stator material and more some stuff are parameters that needs to part of the equations to get decent accuracy.

To do this we need to begin with a known stator and rotor geometry and brake it down to 1 slot/pole calculations. After that scale it back to a complete motor.

/GÃƒÂ¶ran

Goethe
10 W

Posts: 75
Joined: Tue Dec 08, 2009 7:33 pm
Location: In Sweden with polar bears and Santa as neighbours

### Re: Doing the Math

Actually, the whole time I have been working with coreless to K.I.S.S. My next motor is in fact Ã¢â‚¬â€œ a coreless design, so with regards to laminations and eddy currents, we are still on track.

The magnets are Neodymium, the airgap is 1mm between the magnets and the copper, and the Flux Density is 0.5T. I havenÃ¢â‚¬â„¢t mucked with that data yetÃ¢â‚¬Â¦ soon though.

If we can get through the basic calculation so I understands it from beginning to end, that would help me tremendously.

Make sense? KF
* My 2WD Garden Wall
* Current ride: 2WD Disc EBikeKit (9C 2806-equivalent) / Dual Lyen 12FET / 15S6P LiPo when commuting.
* Going to California: 2011: Trip completed
* Club Member: 40-mph & 101. 10k-Club: 10,478 miles-to-date, 4,988 as 2WD.

It is by caffeine alone I set my mind in motion.
It is by the beans of Java that thoughts acquire speed.
The hands acquire shakes, the shakes become a warning.
It is by caffeine alone I set my mind in motion.

Kingfish
1.21 GW

Posts: 3867
Joined: Wed Feb 03, 2010 11:23 am
Location: Redmond, WA, USA, Earth, Sol, Orion–Cygnus Arm, Milky Way. Age: > yesterday < tomorrow

### Re: Doing the Math

Ahaa!

Sorry didnÂ´t mean to get you off track!

So you already have a coreless motor? Or are you designing one?
Is the example with a flat spiral coil moving through a uniform magnetic field close to the real design?
And what is K.I.S.S?

/GÃƒÂ¶ran

Goethe
10 W

Posts: 75
Joined: Tue Dec 08, 2009 7:33 pm
Location: In Sweden with polar bears and Santa as neighbours

### Re: Doing the Math

OK, I suppose I should come cleanÃ¢â‚¬Â¦

I am trying to design the simplest motor I could think of. Powerful Ã¢â‚¬â€œ YES! But still very simple to assemble if one has the correct tooling.

I already have a Radial Flux re-work/modification design completed: I had been working on it for about 3 months and it was going to use a pair of 9C motors to craft my 2WD ebike. But then I learned about other ways to employ electromotive forces by means of an Axial Flux design. In this manner, the magnets are collocated on the sides of the rotor rather than at the circumference. Thus we can theoretically maintain a 1 mm airgap.
In order to assemble the two sides together, a ironless-coreless design is required, which has the added benefit of eliminating cogging.

The magnets under consideration were at one time the very same used in windmills, thus the ideal model would have 16 arc magnets 6.35 mm thick x 100 mm ID x 200 mm OD. The problem with that design is that it requires a steel backing plate to capture the backside flux. I donÃ¢â‚¬â„¢t want to go down that path. Instead, I am going to use a Halbach array so that I can stay with a lightweight aluminum rotor.

This whole time you and I have been working with mock-data; the final numbers will get sorted out as we narrow down the issues. The biggest one I have faced is calculating the windings. I havenÃ¢â‚¬â„¢t found useful tools for figuring that part out Ã¢â‚¬â€œ so I created them.

My goal is to automate the design process; take the current winding application and add several layers to it so one could literally walk through a wizard and design their motor in one night. It can only be done with coreless axial design through because every other design uses radial or iron and that Ã¢â‚¬â€œ as we know makes the calculations more complicated.

K.I.S.S. means Ã¢â‚¬Å“Keep It Simple SillyÃ¢â‚¬Â. When I was in the Navy it meant Ã¢â‚¬Å“Keep It Simple StupidÃ¢â‚¬Â Ã¢â‚¬Ëœcuz most of the swabbies had about an 8th-grade level education. The KISS method works for a lot of things. Sometimes people I work with get way too complicated and it's not necessary to complete the task. We're using KISS here for problem-solving

Are you still on board? ItÃ¢â‚¬â„¢s going to get fun Ã¢â‚¬â€œ I can promise you that!

So to re-cap and outline the challenge:
1. We have a wheel, it spins at a velocity, using x-amount of horsepower.
2. We determined the length of wire, and the amount of current and voltage to theoretically balance the power equation.
3. The next step is to predict how to reduce the current required if we add more magnets and switch to 3-Phase. There are still no losses, or wind resistance, or bearing friction: ItÃ¢â‚¬â„¢s just pure electromotive force that we are attempting to tame into a small compact coreless design.

Make sense?
Truly, I appreciate your help! KF
* My 2WD Garden Wall
* Current ride: 2WD Disc EBikeKit (9C 2806-equivalent) / Dual Lyen 12FET / 15S6P LiPo when commuting.
* Going to California: 2011: Trip completed
* Club Member: 40-mph & 101. 10k-Club: 10,478 miles-to-date, 4,988 as 2WD.

It is by caffeine alone I set my mind in motion.
It is by the beans of Java that thoughts acquire speed.
The hands acquire shakes, the shakes become a warning.
It is by caffeine alone I set my mind in motion.

Kingfish
1.21 GW

Posts: 3867
Joined: Wed Feb 03, 2010 11:23 am
Location: Redmond, WA, USA, Earth, Sol, Orion–Cygnus Arm, Milky Way. Age: > yesterday < tomorrow

### Re: Doing the Math

Ok now it make more sense!

This is really a cool project!

I will reset my brain with some sleep now, and hopefully I have forgotten all about iron stator and rotors and radial flux tomorrow.

/GÃƒÂ¶ran

Goethe
10 W

Posts: 75
Joined: Tue Dec 08, 2009 7:33 pm
Location: In Sweden with polar bears and Santa as neighbours

### Re: Doing the Math

You should keep in mind that, for a coreless design, the airgap is the distance separating the two magnet rotors. So, its not just the magnet-coil distance that counts but the thickness of the coil as well.

How do you plan to construct the axial-flux Halbach array? I have not seen any edge-magnetized wedge magnets (other than custom pieces). Are you just going to use rectangular magnets? If so, the pie-shaped gaps will also count towards the air gap. More importantly, the flux density will vary across the radius - highest on the inside where the magnets touch and lower toward the outside as the gap increases. Not that it wouldn't work, but it would probably work better if you filled the gaps in between with a ferrous material.
Eric

Trek FX 7.3 hybrid - Cyclone 650W setup in process! Still...

rhitee05
10 kW

Posts: 842
Joined: Tue Apr 07, 2009 9:08 pm
Location: Upstate SC

### Re: Doing the Math

EDIT: Appending TOC.

Axial Flux Design, Halbach, Ironless-Stator, Axial Motor

AirGap: On a single-rotor and single stator, it is measured from the face of the magnet-array to the face of the copper winding according to the book that I have on the subject. However with a double-rotor we get the added benefit of both sides and the calcs are a little different. The two sides will be locked N-S in attraction which will make assembly challenging, though less so if I had used an iron stator. My plan was to keep the gap between the two magnetic faces <= 10mm.

At the moment I wish to use Litz wire. I have pre-built a reference to sources here: Magnetic Wire & Nd Magnets Reference

The Archimedes-Winding Calculator is also specific to the AF design.

There is to be an airgap between the windings for cooling should it come to that. The plan is to keep the ohmic heating down to the minimum - though common sense says there will be heat regardless. I might as well be prepared to deal with it.

The magnets have an airgap as well for the same reason - to foment cooling. In the image above I had considered briefly to use 1/4-inch cubes, however custom-orders as a rectangle are possible and will make assembly much easier. I haven't inquired if the manufacturer will do weggies - that would be the optimum!

The construction, if I stay with rectangular shapes, is to either build modular units of four, or to have a jig where I lay down 1/4 of the magnets at a time against the rotor surface (probably the latter). I did not consider putting ferrous material in the bonding agent for use between the gaps; nothing I've read has that as an option. Might be a good subject for FEA. I understand that the Flux Density is greatest where they touch (ID). One positive aspect is that as the inside radius becomes larger the gap between the magnet array reduces. The Halbach array increases the Flux Density by ~1.4X; the whole business is funny math and took many afternoon naps for me to understand it

My hope is that there will be plenty of Torque (LFP has suggested as much) - this way I can focus the design towards light-weight, high-rotation, and efficiency. It's for my next ebike which is structurally stronger and designed to go faster.

Eric, want to help with the maths too? I'll be your friend

Best, KF
Last edited by Kingfish on Fri Mar 04, 2011 2:47 pm, edited 2 times in total.
* My 2WD Garden Wall
* Current ride: 2WD Disc EBikeKit (9C 2806-equivalent) / Dual Lyen 12FET / 15S6P LiPo when commuting.
* Going to California: 2011: Trip completed
* Club Member: 40-mph & 101. 10k-Club: 10,478 miles-to-date, 4,988 as 2WD.

It is by caffeine alone I set my mind in motion.
It is by the beans of Java that thoughts acquire speed.
The hands acquire shakes, the shakes become a warning.
It is by caffeine alone I set my mind in motion.

Kingfish
1.21 GW

Posts: 3867
Joined: Wed Feb 03, 2010 11:23 am
Location: Redmond, WA, USA, Earth, Sol, Orion–Cygnus Arm, Milky Way. Age: > yesterday < tomorrow

### Re: Doing the Math

When discussing airgap, flux density, etc. it's helpful to understand that magnetic flux forms closed circuits just as electric current does, and we can analyze it using the same principles. Magnets behave like voltage sources, the permeability of materials is analogous to conductance (inverse of resistance), and magnetic flux becomes analogous to current.

Let's consider the electric analogy. Take a simple circuit with a source and two resistors in series, one 1k-ohm and one 1-ohm, for a total resistance of 1.001k. It should be obvious that the 1k resistor determines how much current will flow while the 1-ohm resistor has relatively little effect. In a magnetic circuit, iron or another magnetic material has a much, much higher permeability than air. So, even for a very small airgap, the permeability of the air dominates and determines how much flux will flow. Copper is not a magnetic material, so as far as the flux is concerned it's no different than air. Flux requires a closed circuit, so the total airgap around the circuit is what matters.

In your design, the wider airgaps around the outside will increase the magnetic "resistance" there. Just as it would in an electric circuit, that will cause the flux to concentrate toward the inside. The total amount of flux will probably be lower as well. Placing wedges of magnetic material in the gaps between magnets will help with this.
Eric

Trek FX 7.3 hybrid - Cyclone 650W setup in process! Still...

rhitee05
10 kW

Posts: 842
Joined: Tue Apr 07, 2009 9:08 pm
Location: Upstate SC

### Re: Doing the Math

rhitee05 wrote:When discussing airgap, <snip>...
... Placing wedges of magnetic material in the gaps between magnets will help with this.

The idea you tender has merit. My only concern is one of cogging. However since the iron (or similar material) is in the path of the magnets and not in the stator, should we presume the affect is nil? Maybe we should add adhesives & fillers to the Magnetic Wire & Nd Magnets Reference

Best, KF
* My 2WD Garden Wall
* Current ride: 2WD Disc EBikeKit (9C 2806-equivalent) / Dual Lyen 12FET / 15S6P LiPo when commuting.
* Going to California: 2011: Trip completed
* Club Member: 40-mph & 101. 10k-Club: 10,478 miles-to-date, 4,988 as 2WD.

It is by caffeine alone I set my mind in motion.
It is by the beans of Java that thoughts acquire speed.
The hands acquire shakes, the shakes become a warning.
It is by caffeine alone I set my mind in motion.

Kingfish
1.21 GW

Posts: 3867
Joined: Wed Feb 03, 2010 11:23 am
Location: Redmond, WA, USA, Earth, Sol, Orion–Cygnus Arm, Milky Way. Age: > yesterday < tomorrow

### Re: Doing the Math

Adding filler material to the rotors would not cause cogging, and I do not believe it would add eddy losses either.

Cogging is caused by magnetic interaction between the rotor and stator. If the stator is made of entirely non-magnetic materials, as in a coreless motor, no cogging can occur regardless of the construction of the rotor(s).
Eric

Trek FX 7.3 hybrid - Cyclone 650W setup in process! Still...

rhitee05
10 kW

Posts: 842
Joined: Tue Apr 07, 2009 9:08 pm
Location: Upstate SC

### Re: Doing the Math

Right then, IÃ¢â‚¬â„¢m on the hunt for bonding agents and magnetic fillers this week. If you good lads should find one of interest IÃ¢â‚¬â„¢d be much obliged if you tossed the hot potato my ways please.

Hmm, silly question: With regards to my post on Fri Sep 10, 2010 4:59 pm, I calculated Inductance (L) to be 26.43 ÃŽÂ¼H. Is that math correct? Admittedly it hasnÃ¢â‚¬â„¢t caused me to lose sleep, nonetheless I am a bit concerned about accuracy. Pesky thing that Ã¢â‚¬â€œ accuracy.

Cheers, KF
* My 2WD Garden Wall
* Current ride: 2WD Disc EBikeKit (9C 2806-equivalent) / Dual Lyen 12FET / 15S6P LiPo when commuting.
* Going to California: 2011: Trip completed
* Club Member: 40-mph & 101. 10k-Club: 10,478 miles-to-date, 4,988 as 2WD.

It is by caffeine alone I set my mind in motion.
It is by the beans of Java that thoughts acquire speed.
The hands acquire shakes, the shakes become a warning.
It is by caffeine alone I set my mind in motion.

Kingfish
1.21 GW

Posts: 3867
Joined: Wed Feb 03, 2010 11:23 am
Location: Redmond, WA, USA, Earth, Sol, Orion–Cygnus Arm, Milky Way. Age: > yesterday < tomorrow

### Re: Doing the Math

Kingfish wrote:Admittedly it hasnÃ¢â‚¬â„¢t caused me to lose sleep, nonetheless I am a bit concerned about accuracy. Pesky thing that Ã¢â‚¬â€œ accuracy.

I don't want to crush all of your hopes and dreams, but in this case accuracy is the realm of FEM tools or a ridiculous amount of time with equations and integrals. We really only have equations for the most trivial of geometries, and even those are likely to be upset by the collision with reality to some degree. But, since you asked...

<deep breath>

Starting back with the calculation that you need 2.53m of wire for X torque, assuming Y current and Z flux. That's all well and good, but that's only valid for 2.53m of wire in the radial direction. The equation F = IL x B is a vector equation, and the directions of L and B determine the direction of F. In order to produce a force that is entirely in the tangential direction, L must be in the radial direction (assuming B is perpendicular to the surface). If L is offset from the radial direction, then only a portion of the resulting force will the in the useful tangential direction and the rest will be in the useless radial direction.

The circular geometry of your example spiral winding is actually easy to analyze. It so happens that the "effective" length of each half-circle of wire is 2*r, so the effective length of the coil is the actual length multiplied by 2/pi. To get the desired effective length of 2.53m would actually require a coil of about 4m of wire. Only about 64% of the wire contributes to torque, but it all experiences loss. The exact percentage varies depending on geometry but it's impossible to get 100%.

Regarding the inductance, the spiral happens to be one of the simple geometries that we do have equations for. However that will not be the same as the wedge-shaped coils shown in your picture. Even if you did find a way to calculate the open-air inductance of that coil shape, it would still be different in situ because of the presence of the permanent magnets and magnetic materials. It really is almost impossible to make an accurate calculation, there are just too many factors which will affect the inductance of the physical coil.
Eric

Trek FX 7.3 hybrid - Cyclone 650W setup in process! Still...

rhitee05
10 kW

Posts: 842
Joined: Tue Apr 07, 2009 9:08 pm
Location: Upstate SC

### Re: Doing the Math

Ahh well, my Cabernets are crushed, but not my spirit! So Ã¢â‚¬â€œ moving right along, this means that we will have to be rigorous with our deductive testing

I might as well design from the outset with planned losses. Let me step back a moment then to gather new examples to study, for if the concept can be understood I will be better for it than hunting for boar guilelessly armed with a blunt stick

There are several questions now and I shall try to keep this organized; the proverbial can oÃ¢â‚¬â„¢ worms has been kicked:

Air Gaps and Filler:
Briefly I investigated what I think might be a more suitable approach than an initial idea of some additive to the bonding agent.
• Soft Core material: Ignored.
• Electric Steel Laminations: 4000X that of air. Good choice and commonly available even locally in my metro.
• Permalloy: Twice that of Electric Steel. No sources.
• Mu-metal: 5X that of Electric Steel. Expensive, though has my imagination twitching wildly.
This now brings up a question of construction, and methinks the most economical would be to have wedges that fit between the bar magnets rather than a challenging water-jet-cut cage. IÃ¢â‚¬â„¢m just thinking out loud; the wedges would be very cheap to fabricate, less waste of material, and better than a poke in the eye Ã¢â‚¬â€œ as in doing nothing. See Figure 1.

(Green) Electric Steel Lams between magnets

Magnetic Circuit:
Similarly, I got to thinking about completing the magnetic circuit as you well pointed out. On the left side of Figure 2 is a representation of the initial assembly: Two rotors with magnets separated by a cap at the top and bearings at the bottom (not shown), with a stator in the center and copper in-between.

The second idea to explore is using a horseshoe shape for routing the flux as displayed on the right of Figure 2. Fanciful, though I wonder how practical. Moving onÃ¢â‚¬Â¦

Double-Sided Rotor & Stator arrangements

Simplistically weÃ¢â‚¬â„¢re taught that the magnetic force of attraction is inversely proportional to the square of the distance. However in a double-sided Halbach arrangement, the calculations are expressed here:

I bought the book, but found the references on Google first. Actually the section begins on Page 107, with the Equation for Peak Flux at the top of Page 109 (3.42), and in the double-sided configuration with 3.45 & 3.46.

Bm0 = Br[1 - exp(-ÃŽÂ²hM)]((sin(Ãâ‚¬/nM))/(Ãâ‚¬/nM)) [3.42]
Solve for Bm0, where
hM = 6.35mm (1/4-inch)
nM = 4
ÃŽÂ² = 2Ãâ‚¬/la,
la = nM * (hM+ Ag) where Ag = median airgap between magnets = 10mm. This figure will vary depending on the ID of the rotor face and the number of poles in the array.
If I use an N52 cube magnet, Br = 1.48
Therefore, Bm0 = 0.61 T
We can use 0.61 T if you like, but I rounded down to 0.5 T for the sake of discussion.

The part where IÃ¢â‚¬â„¢m a little bit lost is ÃŽÂ²x and ÃŽÂ²z, however I have space between the discs (z) as 10 mm (again for the sake of discussion).

IÃ¢â‚¬â„¢m going to hold on the Winding Calculation until we sort out the Flux Density.

Have I lost anyone?
~KF
Last edited by Kingfish on Fri Mar 04, 2011 2:49 pm, edited 1 time in total.
* My 2WD Garden Wall
* Current ride: 2WD Disc EBikeKit (9C 2806-equivalent) / Dual Lyen 12FET / 15S6P LiPo when commuting.
* Going to California: 2011: Trip completed
* Club Member: 40-mph & 101. 10k-Club: 10,478 miles-to-date, 4,988 as 2WD.

It is by caffeine alone I set my mind in motion.
It is by the beans of Java that thoughts acquire speed.
The hands acquire shakes, the shakes become a warning.
It is by caffeine alone I set my mind in motion.

Kingfish
1.21 GW

Posts: 3867
Joined: Wed Feb 03, 2010 11:23 am
Location: Redmond, WA, USA, Earth, Sol, Orion–Cygnus Arm, Milky Way. Age: > yesterday < tomorrow

### Re: Doing the Math

Don't take anything I said to mean that there isn't value in analyzing a simple case like the one you used. Just have realistic expectations. Also, even a not-quite-right expression will (usually) give a result that's in the same ballpark as reality. That can be instructive, too!

Kingfish wrote:This now brings up a question of construction, and methinks the most economical would be to have wedges that fit between the bar magnets rather than a challenging water-jet-cut cage. IÃ¢â‚¬â„¢m just thinking out loud; the wedges would be very cheap to fabricate, less waste of material, and better than a poke in the eye Ã¢â‚¬â€œ as in doing nothing. See Figure 1.

Wedges sound reasonable. You're along way from construction, but a word of magnet caution - those wedges will be powerfully attracted as you try to put them in place! I wouldn't worry too much about using a high-end magnetic material, I doubt you would be able to tell the difference. The airgap will still dominate the magnetic circuit. At this point I feel it does beg the question - at what point does the effort of making the Halbach array make it worthwhile to do the simpler wedge magnets and flux ring instead?

Kingfish wrote:The second idea to explore is using a horseshoe shape for routing the flux as displayed on the right of Figure 2.

You're getting the idea, which is good, but this is not necessary and would actually be counterproductive! The complete magnetic circuit in this case does not run through the air around the stator, but instead through adjacent pairs of poles on the two rotors. That's the function of either the flux ring or Halbach array. A little flux bridge like you've drawn, if it had any effect, would likely serve to divert some of the useful flux around the stator. You're thinking in the right direction, though.

Your book looks like a good reference. In this case Bx and Bz refer to the two components of the flux vector. Bz is the useful flux perpendicular to the magnet faces. Bx is wasted flux parallel to the faces. The equations express them as functions of x, moving horizontally across the array, and z moving from one rotor to the other. Probably the most useful value for you would be the value of Bz centered between a pair of poles. In that case, z=t/2, so the two cosh terms will cancel and the sine will be 1 at x=0, so Bz=Bm0.
Eric

Trek FX 7.3 hybrid - Cyclone 650W setup in process! Still...

rhitee05
10 kW

Posts: 842
Joined: Tue Apr 07, 2009 9:08 pm
Location: Upstate SC

### Re: Doing the Math

Probably the most useful value for you would be the value of Bz centered between a pair of poles. In that case, z=t/2, so the two cosh terms will cancel and the sine will be 1 at x=0, so Bz=Bm0.

I am very happy to learn that. One source in particular suggested the best transformer would be one where the wires were embedded with the laminations (not possible, but still the best one could hope for). Indeed, I was hoping that the flux density between the rotors would be similar in theory. I suspect that it would only grow stronger should hM, Br increase, or z decrease Ã¢â‚¬â€œ all of which is possible.

Flux Ring:
I presume you mean this sort of arrangement in Figure 3, yes?

That design requires a backing steel plate, or at least some way to capture the flux on the backside, maybe with a sheet of electric steel. Also there are only 8 pole-pairs, meaning high-rotation. LetÃ¢â‚¬â„¢s compare/contrast with for example a 9C 2806 hub, which is what I presently own Ã¢â‚¬â€œ and I should add that I find it to be quite a capable motor and a worthy challenge to betterÃ¢â‚¬Â¦

• 9C 2806: 23 pole-pairs, 51 Teeth. Magnets are 3 x 13.5 x 27 mm. K = 0.93 (according to ebikes.ca). I would be keen to know what the Tesla rating is for these magnets; for argument I presume that are 42N since that is commonly available, though perhaps the temp rating is higher.
Ref: http://www.ebikes.ca/simulator/
If you enter the custom values as 62V and 32A that is just about what I am achieving. On a flat at WOT, depending on wind I top out between 33-36 mph, and obviously faster on a downgrade. ItÃ¢â‚¬â„¢s plenty quick for urban, however I want and desire to go faster yet.

• Windmill: magnets Â¼-in x 4-in ID x 8-in OD, N42 is common, and Â½-in is available. 8 pole-pairs; about 1/3 the gearing of the 9C. Even with double-sided, I fear the rotation is much too fast. Definitely the easiest to assemble having the fewest poles and essentially no air gap between the adjacent poles. See Figure 3.

• Halbach Array: magnets 6 x 25 x 10 mm x 2 = one pole, with 20 pole-pairs & 21 teeth; that's 43% of the poles for the 9C 2806, and more than twice the Windmill. The airgap between adjacent magnets is 4mm at the OD. Planned Coil cross-section is 36 mm^2 within the 10 mm air gap (z). Leaves me room to add 1 mm carbon fiber facing/reinforcement on each side to retain the windings to the stator plate. See Figure 4.
EDIT: Corrected typo.

Horseshoe Revisited:
A little flux bridge like you've drawn, if it had any effect, would likely serve to divert some of the useful flux around the stator. You're thinking in the right direction, though.

Ha! Now IÃ¢â‚¬â„¢ve got the Ã¢â‚¬Å“happy puppyÃ¢â‚¬Â syndrome Ã¢â‚¬â€œ with my tail all wagging. Early on I had considered adding a Radial Flux component to the design but discarded it because of the issue of trying to reduce the airgap between the winding and the magnetic face. In reference with Figure 5, is that what you meant?

Miles of smiles, KF
Last edited by Kingfish on Sun Mar 06, 2011 3:40 pm, edited 2 times in total.
* My 2WD Garden Wall
* Current ride: 2WD Disc EBikeKit (9C 2806-equivalent) / Dual Lyen 12FET / 15S6P LiPo when commuting.
* Going to California: 2011: Trip completed
* Club Member: 40-mph & 101. 10k-Club: 10,478 miles-to-date, 4,988 as 2WD.

It is by caffeine alone I set my mind in motion.
It is by the beans of Java that thoughts acquire speed.
The hands acquire shakes, the shakes become a warning.
It is by caffeine alone I set my mind in motion.

Kingfish
1.21 GW

Posts: 3867
Joined: Wed Feb 03, 2010 11:23 am
Location: Redmond, WA, USA, Earth, Sol, Orion–Cygnus Arm, Milky Way. Age: > yesterday < tomorrow

### Re: Doing the Math

Kingfish wrote:Flux Ring:
I presume you mean this sort of arrangement in Figure 3, yes?

That's one configuration, but not the only one possible. With the flux ring, it's no longer necessary (or actually desirable) to have the magnets tight against each other. The magnetic circuit is completed through the flux ring rather than directly from magnet to magnet. If the magnets are too close together, some of the flux will actually short-circuit to an adjacent magnet rather than across the stator. I believe the usual rule of thumb is to keep the separation equal or greater than the airgap. Since separation isn't an issue, you can use any convenient shape of magnet. You could make a large diameter, high pole-count rotor out of circular puck magnets, for example.

Kingfish wrote:Ha! Now IÃ¢â‚¬â„¢ve got the Ã¢â‚¬Å“happy puppyÃ¢â‚¬Â syndrome Ã¢â‚¬â€œ with my tail all wagging. Early on I had considered adding a Radial Flux component to the design but discarded it because of the issue of trying to reduce the airgap between the winding and the magnetic face. In reference with Figure 5, is that what you meant?

Er, you were on the right track but you must've taken a hard right turn while I wasn't looking.

For the axial flux design, the only useful flux is exactly that - in the axial direction directly from rotor to rotor. Are you familiar with vector cross-products? (For the moment I'll assume you're not, otherwise feel free to ignore me) Going back to your Lorentz equation, the direction of the force is determined by L x B, that is the direction of the current cross the direction of B.

First of all, we can easily show that the current flowing through the end windings (that is, tangential around the circumference) will never give us a useful result. A tangential L vector (theta-directed in engineering speak) can never produce a force that is also in the theta-direction. It could only produce a force in the radial (r) direction or axial (z) direction. Both of those do nothing but put stress on the stator.

Now let's look at the current in the radial direction. A flux in the radial direction will do nothing - r x r = 0. A flux in the theta-direction will produce force in the +z or -z, doing no useful work except stressing the stator. Finally, z-directed axial flux produces theta-directed force and thus useful torque. So, any flux that does not a)travel in the z-direction and b) pass through the stator windings does not do any useful work for us.

Radial flux such as you've drawn would only do useful work if current is flowing in the axial direction... would would be a standard radial-flux motor! I'm sure some sort of hybrid is possible, but I doubt that's a rabbit hole you want to go down.
Eric

Trek FX 7.3 hybrid - Cyclone 650W setup in process! Still...

rhitee05
10 kW

Posts: 842
Joined: Tue Apr 07, 2009 9:08 pm
Location: Upstate SC

### Re: Doing the Math

Er, you were on the right track but you must've taken a hard right turn while I wasn't looking.

Gotcha. ThatÃ¢â‚¬â„¢s why I discarded the concept early on, but for a moment you had me thinking of something else. Was it perhaps the laminate shoes as indicated in Figure 6 as another type of Flux Ring? Hmmm, I prefer to keep it simple.

Air Gaps & Adjacent Magnet Spacing:
It seems to me that the Air gap between Magnets (Agw) and the surface of the Copper Windings should be closest. Are you suggesting that the distance between magnetic faces (z) should also be less than the air gap between adjacent magnets (Agm) as well so as to promote the inductance across z (Agz)?

With all these air gaps, I figure we better have some shorthand

BTW Ã¢â‚¬â€œ thanks for explaining this
~KF
Last edited by Kingfish on Fri Mar 04, 2011 2:51 pm, edited 1 time in total.
* My 2WD Garden Wall
* Current ride: 2WD Disc EBikeKit (9C 2806-equivalent) / Dual Lyen 12FET / 15S6P LiPo when commuting.
* Going to California: 2011: Trip completed
* Club Member: 40-mph & 101. 10k-Club: 10,478 miles-to-date, 4,988 as 2WD.

It is by caffeine alone I set my mind in motion.
It is by the beans of Java that thoughts acquire speed.
The hands acquire shakes, the shakes become a warning.
It is by caffeine alone I set my mind in motion.

Kingfish
1.21 GW

Posts: 3867
Joined: Wed Feb 03, 2010 11:23 am
Location: Redmond, WA, USA, Earth, Sol, Orion–Cygnus Arm, Milky Way. Age: > yesterday < tomorrow

### Re: Doing the Math

Kingfish wrote:Was it perhaps the laminate shoes as indicated in Figure 6 as another type of Flux Ring? Hmmm, I prefer to keep it simple.

That is a valid configuration. Not sure about the relative merits, but it would work.

Kingfish wrote:Air Gaps & Adjacent Magnet Spacing:
It seems to me that the Air gap between Magnets (Agw) and the surface of the Copper Windings should be closest. Are you suggesting that the distance between magnetic faces (z) should also be less than the air gap between adjacent magnets (Agm) as well so as to promote the inductance across z (Agz)?

With all these air gaps, I figure we better have some shorthand

Sorry, I'm probably being a little unclear in my terminology. When I say airgap in this context (a coreless axial flux machine) I'm referring to the entire distance from rotor to rotor. A better term is probably "rotor spacing", which would reserve the term "airgap" for the usual magnet-to-stator definition.

The important concept here is that flux will take the path of least resistance (like electric current, water, etc). Remember that all the stator materials here (copper, aluminum, carbon fiber, whatever) all equivalent to air in terms of reluctance, so for now let's pretend the stator is gone. All the flux cares about is going from N of one magnet, to S of another magnet, then from N of the 2nd magnet back to S of the first to form a complete circuit. It'll take whatever the shortest (least reluctance) path is to do so. If one of the adjacent magnets is closer than the opposite rotor magnet, some of the flux will tend to take that path instead of the useful one through the stator.

This isn't true of the adjacent magnets in the Halbach configuration, since they are oriented to form the desired flux path. You would want the opposite rotor pole to be closer than the next Halbach pole, though, for the same reasons as above.

For fun, a couple of FEM examples:
Planar Array1.png
Magnet-to-magnet spacing 2x airgap distance

Planar Array2.png
Magnet-to-magnet spacing 1/2 airgap distance
Eric

Trek FX 7.3 hybrid - Cyclone 650W setup in process! Still...

rhitee05
10 kW

Posts: 842
Joined: Tue Apr 07, 2009 9:08 pm
Location: Upstate SC

### Re: Doing the Math

Understood. Neat FEA pics; did you make those?

OK - So image me as a dog with a bone; IÃ¢â‚¬â„¢m not quite finished chewing on this yet:

In a Halbach Array, the orientation will be

Ã¢â‚¬Â¦L-S-R-N-L-S-R-NÃ¢â‚¬Â¦
With L = Left, S = South, R = Right, and N = North, repeating etc.

In my design I can guarantee that the distance between a N and a S pole at the shortest leg on the ID of the Rotor face will be just about twice as far as the z-distance, the Agz, the opposite rotor pole. Therefore I think we are good here and in the pink. <nods>

In this case, does it matter then if in the Halbach array the adjacent magnets touch since we want the L and R flux to be directed towards N and away from S? See Figure 7.

I have another flux ring for you since weÃ¢â‚¬â„¢re talking out loud:
The top diagram in Figure 7 is a Halbach Array representation: Nice nÃ¢â‚¬â„¢ happy flux is flowing, we have a dandy lilÃ¢â‚¬â„¢ sine-wave, looks pretty good, all is swell.

The bottom image has pole shoes applied. Now, IÃ¢â‚¬â„¢ve seen a diagram like this on the K&J Magnetics website and itÃ¢â‚¬â„¢s also in the book IÃ¢â‚¬â„¢ve been using Ã¢â‚¬â€œ though the authors could be talking Swahili as I donÃ¢â‚¬â„¢t understand a word of the technical speak (recoil line, excitation current, permeance, and demagnetization action on pages 99-102). It seems to me that putting shoes on a magnet shorts out the circuit and reduces the force; in fact IÃ¢â‚¬â„¢ve seen horseshoe magnets stored this way with a little bar across to tie the field down. I wouldnÃ¢â‚¬â„¢t think weÃ¢â‚¬â„¢d want something like that in this motor unless maybe I am in the midst of assembly. Am I making any sense?

BTW - So far I havenÃ¢â‚¬â„¢t seen anything negative about the design other than it will be a bugger to secure the magnets.
Best, KF
Last edited by Kingfish on Fri Mar 04, 2011 2:53 pm, edited 1 time in total.
* My 2WD Garden Wall
* Current ride: 2WD Disc EBikeKit (9C 2806-equivalent) / Dual Lyen 12FET / 15S6P LiPo when commuting.
* Going to California: 2011: Trip completed
* Club Member: 40-mph & 101. 10k-Club: 10,478 miles-to-date, 4,988 as 2WD.

It is by caffeine alone I set my mind in motion.
It is by the beans of Java that thoughts acquire speed.
The hands acquire shakes, the shakes become a warning.
It is by caffeine alone I set my mind in motion.

Kingfish
1.21 GW

Posts: 3867
Joined: Wed Feb 03, 2010 11:23 am
Location: Redmond, WA, USA, Earth, Sol, Orion–Cygnus Arm, Milky Way. Age: > yesterday < tomorrow

### Re: Doing the Math

Just your friendly neighborhood tourist droping by to say keep the discusion going.....

I looked at Halbach arrays while playing with motor design.....my biggest issue was the increased pole count...& the issue of all the other magnets not contributing to the motor's power generation. by the time you add all the magnets for the effect, you are close to the same weight as an iron flux ring on my scale models ( I copped out & went low road)

The bigger isue also was flat/square magnets layed out to create the halbach effect on a radius that is in the vein of Axial topography.......I went looking for some custom magnet manufacturer to "zap" up a set with the N/S poles askew 45 or 60 deg. & in a modified "pie" shape(just like the launchpoint) but I am a lowly basment experimentor...with nothing but old ideas & bad habits to create with....so I side stepped(gave up untill I can manufacture my own magnets) & moved onto utilizing tangibles within my reach.

Given my status as a confirmed "dullard", I will be watching intently from the shawdows for some scraps of knowledge to fall within my reach. Where in I will pounch on it & incorperate into my database, & re-spew it in a build that will make me look far schmarter'r then I really am.
Thanks for this thread KF.
I assumed you have seen this thread KF, but never assume as a they say.
viewtopic.php?f=28&t=13957&hilit=axial+flux+discusion
get some......

All information & advice provided by Thud are "Open Source" & free for personal use & distribution under the following agreement linked below.

Thud
100 MW

Posts: 2518
Joined: Thu Aug 20, 2009 6:20 am
Location: West Michigan,USA

### Re: Doing the Math

KF, have you thought of a stator solution with PCB's ? The one on the picture is 780 Âµm thick and contains 140 gr copper. They can be stacked in unlimeted number.

/GÃƒÂ¶ran

Goethe
10 W

Posts: 75
Joined: Tue Dec 08, 2009 7:33 pm
Location: In Sweden with polar bears and Santa as neighbours

### Re: Doing the Math

After sleeping on the air gap between adjacent magnets (Agm), I would like to add another diagram in hopes of crystalizing the issue for better comprehension:

In Figure 8 we have two Magnetic Arrays; if these are of Halbach design, the flux density (ÃÂ¤) on the left will vary and weaken as the radius from the center expands, while the one on the right should remain consistent - at least in the ideal sense. We can likewise observe that the air gap widens on the left (left dimension), whilst the air gap on the right is consistent.

Conclusions:
• The Left array could benefit if the air gap was filled with more permeable material such as electric steel because it would condition and spread the field strength as the radius expands towards the OD. See Figure 1.
• The Right array does not require filler material since the geometry already propagates the desired affects.
• If the Left array requires filler material, does this material need to touch the adjacent magnets?
• What is a reasonable ratio of air gap to adjacent magnet?

I think once we get this resolved we can return to the windings aspect.

Thud: Oh gosh, IÃ¢â‚¬â„¢ve read and re-read your posts on the subject, and searched and plumbed the depths of ES on various aspects in the quest to comprehend this quirky twisty AF solution. The question I have for you is:
• When you asked for bids on the custom design of the shape, were you daunted by the quote or quantity? Can you briefly outline the issues that you faced? Perhaps together as a team we can overcome this, yes?

Goethe: Yes, a PCB stator has been conceived as one way to accomplish the windings. LFP has the same recommendation. We need to evaluate the characteristics of the windings relative to the flux density and the expected torque before evaluating which is the best approach to packaging. BTW Ã¢â‚¬â€œ that is an awesome pic of a PCB stator! Where did you get that? I can easily imagine a multilayer heavy-copper board. Can you imagine though the torque upon the trace face when going WOT? I wonder how thin we could make an internal layer?

Winding Material:
OK Ã¢â‚¬â€œ IÃ¢â‚¬â„¢m taking the plunge. Have you guys considered the multipath Litz wire to reduce eddy currents? I am imagining the Hz becoming something like 7 rps * Teeth/Phase-Count and then multiply that by how much faster than [30 mph/ 48,3 km/h] we wish to go. ThereforeÃ¢â‚¬Â¦

Solve for frequency (f) if the number of poles-pairs is 20, and the number of teeth is 21 for a 3-phase motor at 7 rps:
f = (21 teeth / 3 phase) * 7 rps = 7 * 7 = 49 Hz (previously solved on Sat Sep 11, 2010 12:38 pm).
If I want to go 1.5X faster then f would need to be 1.5 * 49 = 73.5 Hz. Nothing here suggests fear of eddy currents from high frequency. The part that is missing is how fast we need the microcontroller to pulse the FET stage, and I donÃ¢â‚¬â„¢t have an answer to that.

Regardless, I think itÃ¢â‚¬â„¢s worth evaluating all the winding options, and to do that we need to calculate the turns per winding next.

Best, KF
Last edited by Kingfish on Fri Mar 04, 2011 2:53 pm, edited 1 time in total.
* My 2WD Garden Wall
* Current ride: 2WD Disc EBikeKit (9C 2806-equivalent) / Dual Lyen 12FET / 15S6P LiPo when commuting.
* Going to California: 2011: Trip completed
* Club Member: 40-mph & 101. 10k-Club: 10,478 miles-to-date, 4,988 as 2WD.

It is by caffeine alone I set my mind in motion.
It is by the beans of Java that thoughts acquire speed.
The hands acquire shakes, the shakes become a warning.
It is by caffeine alone I set my mind in motion.

Kingfish
1.21 GW

Posts: 3867
Joined: Wed Feb 03, 2010 11:23 am
Location: Redmond, WA, USA, Earth, Sol, Orion–Cygnus Arm, Milky Way. Age: > yesterday < tomorrow

Next