## Doing the Math

Electric Motors and Controllers

### Re: Doing the Math

Another comment about the number of windings. You may already know this, but I'll add it for the benefit of others who might be reading along.

The total number of wires per slot determines the torque/power potential of the motor. Using the terms "slot" or "tooth" is a little imprecise when talking about a coreless motor here, but I think everyone will understand the intended meaning. How those wires are connected then determines the Kv/Kt of the motor. For example, if you made each winding a single coil of 28 turns in series, you would have a motor with very low Kv and high Kt. If you did the other extreme and placed 28 single coils in parallel, you would have a motor with high Kv and low Kt, with a whole spectrum of choices in between.
Eric

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rhitee05
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### Re: Doing the Math

rhitee05 wrote:Okay, that explains the design choice. I'm a little confused by the details however. You say that the magnet OD is 180 mm and the magnets are 30 mm long, which would imply that the ID is 150 mm and the average diameter is 165 mm. But you say that the average (midpoint) diameter is 150 mm?

• Magnets in Plan-A are 30 mm in length.
• Let's start with the 180 mm OD and convert that to r = d/2 = 90 mm.
• Subtract 30 from 90 = 60; 60 mm r = 120 mm ID.
• 1/2 of 30 mm is 15 mm; 60 + 15 = 75; midpoint = 75 mm r or 150 mm diameter.
Make sense?

For the geometry you posted (fig 4), I actually think we can make a big hand-waving simplification and say that the effective length of each coil is 2x the magnet length. Why you ask? Especially for the coreless case, we can assume with a reasonable degree of accuracy that the magnetic field is uniform across the face of each magnet and zero outside the face. There will be some fringing fields, edge effects, etc, but this assumption makes things very easy and is fairly accurate. Since the only portion of the coil of interest is the portion within the magnetic field, and due to the coil shape these sections are entirely in the radial direction, we can simply say that the effective length is 2x the magnetic length (one for each side of the coil). Simple, eh?

Works for me

Another suggestion I'll toss out is to consider using overlapping windings. That will generally achieve higher copper fill (more power density), but at the cost of having more stator poles and thus higher electrical frequency. But, that doesn't seem like it should be a problem here.

Considered overlapping however the Book AF Machines makes a rather persuasive case for non-overlap windings, and above that, non-overlaps are easier to assemble
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### Re: Doing the Math

rhitee05 wrote:Another comment about the number of windings. You may already know this, but I'll add it for the benefit of others who might be reading along.

The total number of wires per slot determines the torque/power potential of the motor. Using the terms "slot" or "tooth" is a little imprecise when talking about a coreless motor here, but I think everyone will understand the intended meaning.

Normally I am good with using slots over teeth; the issue I face is that winding can be confused with windings and that means exactly the same to me as well. We could propose to reserve winding(s) as refering to the turns of copper. A tooth or slot is the same to me in a coreless design, though since there are no teeth or slots we are again left to consider the proper term.

Now another way to look at the whole arrangement is this:

For Plan-A, I intend to use Non-Overlap Windings; that is the term to describe the assembly of copper turns as well as the orientation of the structure. Maybe what we need to do is set the definition:
• Winding: A single unit of copper turns, whether multi-level or multi-row.
• Windings: Plural of Winding, the whole assemblage, or a phase-array thereof.
• Turn: One revolution of copper conduit, whether it be of a single conductor, an assembly of parallel wires, or of Litz wire.
• Turns: Plural of Turn, regardless of quantity or structure or layout.
Definitions can be cool

How those wires are connected then determines the Kv/Kt of the motor. For example, if you made each winding a single coil of 28 turns in series, you would have a motor with very low Kv and high Kt. If you did the other extreme and placed 28 single coils in parallel, you would have a motor with high Kv and low Kt, with a whole spectrum of choices in between.

Oh yes: We are just now getting to the meat and potatoes of the design! Imagine that this thread started on August 31st and I am just now comprehending the ratio of Power to Turns

What I'd like to do next is develop a couple of various design changes and determine the physical results. If the math is proving correct and we can anticipate with confidence the results - then I will be able to programmatically develop an AF Motor wizard to speed our tweaking along, similar to how I have been leveraging the Archimedes Winding Calculator.

Make sense?
~KF
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### Re: Doing the Math

Kingfish wrote:# Magnets in Plan-A are 30 mm in length.
# Let's start with the 180 mm OD and convert that to r = d/2 = 90 mm.
# Subtract 30 from 90 = 60; 60 mm r = 120 mm ID.
# 1/2 of 30 mm is 15 mm; 60 + 15 = 75; midpoint = 75 mm r or 150 mm diameter.

Sigh. I seem to be having problems with math recently. You are, of course, correct. Your use of terminology seems reasonable and fairly standard, so there should be little confusion.
Eric

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rhitee05
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### Re: Doing the Math

rhitee05 wrote:
Kingfish wrote:# Magnets in Plan-A are 30 mm in length.
# Let's start with the 180 mm OD and convert that to r = d/2 = 90 mm.
# Subtract 30 from 90 = 60; 60 mm r = 120 mm ID.
# 1/2 of 30 mm is 15 mm; 60 + 15 = 75; midpoint = 75 mm r or 150 mm diameter.

Sigh. I seem to be having problems with math recently. You are, of course, correct. Your use of terminology seems reasonable and fairly standard, so there should be little confusion.

NO NO!! I make mistakes ...but then I don't mind admitting them

We're in this together, you and I, us and we'all

Let's make some motors! KF
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### Re: Doing the Math

You may have already seen this, but if not I suspect you will find it very useful:

http://web.mit.edu/scolton/www/SCThG.pdf

This is a M.S. thesis by a fellow named Shane Colton at MIT, who I believe lurks around here periodically, on the topic of BLDC motor design and control. The entire document is interesting, but there is one section in particular I think you will find highly relevant. He goes through, in great detail, the design and modeling process for BLDC motors and his particular test case is a coreless axial flux design. He goes through several models with increasing fidelity and ends up with measured data to compare against. Should be a very useful reference for you as you attempt the same task.

Eric
Eric

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rhitee05
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### Re: Doing the Math

I began following Shane's blog last spring; once I completed the elementary motor resign for Radial Flux I switched over to AF based on his discoveries. Cool stuff, huh?

Plan-B
Similar to Plan-A, the length of the Magnets changes from 30 mm to 40, reducing the Torque Arm length by 5 mm.
r = 0.070 m
F = Ãâ€ž / r = 33.9 / 0.070 m = 484 N.
Solve for L:
L = Fx / [(2/Ãâ‚¬)IBz] => 484 /[(2/Ãâ‚¬) *44 * 0.5] = 484 / 14 = 34.6 m (which is equal to 2d)
Solve for length of wire per phase per tooth (Lpp):
Lpp = Lw / 3 / 7 => 34.6 / 3 / 7 = 1.65 m which is also equal to 2d,
1.65 / 2 = .8236 m / 823.6 mm as a single diameter.
With Perimeter determined by CAD; Calculate ID:
Pc = 105.1 mm = 2 Ãâ‚¬ r => r = Pc / (2 Ãâ‚¬) = 105.1 / (2 * Ãâ‚¬) = 16.7 mm
We want diameter; 2 * r = d => 2 * 16.7 = 33.47 mm
A crude estimate of the number of turns per tooth would be 823.6 / 33.47 = 24.6 Turns

Using 4:1 Flat Wire, 20 AWG = 0.39 High x 1.56 wide; the Archimedes Winding Calculator suggests that 24.6 turns with an ID of 16.7 mm = 2.07 m of conductor, and a 36.3 mm OD.

Conclusions:
By lengthening the magnets, though constrained by the OD, the length of the conductor becomes longer, however so does the ID of the winding, and fortunately for us the number of turns is reduced, as is the actual length of copper to create the turns, therefore our resistance likewise is reduced, and overall efficiency is improved.

There is however another benefit yet to be calculated that is shared by both of these Plans. And there are other tweaks we can do to this design.

More in a bit, KF
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### Re: Doing the Math

Magnet Height:
The length of the magnets is not the only dimension that could be changed; increasing the height also helps increase the Tesla value Ã¢â‚¬â€œ but only up to a point. I created a graph - Figure 10, which displays roughly the trend and the point of narrowing gain as Height increases and the Tesla value approaches 100%. The break-even point is about 8 mm +/- 1mm.

Rotor Air Gap:
We can also reduce the air gap along the z-axis between the rotor plate to raise the value of Tesla. In Figure 11 the graph is nearly linear for small distances. By reducing the air gap we increase Tesla. There is however a practical cutoff and that is the width of the windings + the air gap between the winding and the magnet surfaces x 2.

Magnet Mass:
LetÃ¢â‚¬â„¢s consider for a moment that we increase the thickness of the magnets to 8 mm. With Plan-B, the mass for one pole is 105 gm (given to us by the magnet wizard here). Since we are using a Halbach array we will require 40 magnets per side: 40 * 2 * 105 gm = 8400 gm / 8.4 kg / 18.5 lbs. That is a shed-load of rotating mass! And even though it has the potential to raise our Tesla value quite significantly, the cost of manufacturing and the weight factor in pretty large. BTW, Plan-A magnets which are 10 mm shorter and only 6 mm high weigh in at 5.064 kg / 11.2 lbs; that is still quite heavy for the power output in my book and I think we can do much better.

Introspection:
Perhaps we need to re-think this problem. We can change the length and height of the magnets, and distance between the opposing faces to affect Tesla. And weÃ¢â‚¬â„¢re kinda stuck with the radius as well Ã¢â‚¬â€œ at least that is the radius needs to be about on par with the 9C 2806 hub. We actually have a little bit of wiggle room to play here but not too much. For example, we know that as the radius decreases that we need to apply more Force (F), and that in turn leads us down the path of larger more powerful magnets and longer conductors.

What happens though if we go the other way; what if we changed the OD of the magnet ring to 200 mm and reduced the length of the magnets from 30 or 40, down to 20 mm since our torque arm is longer. All things being equal, letÃ¢â‚¬â„¢s do the math and find out...

Plan-C
Identical to Plan-A except for the following:
Magnet length = 20 mm; Â½ length = 10 mm.
r = (200 / 2) Ã¢â‚¬â€œ 10 = 90 mm / 0.090 m.
F = Ãâ€ž / r = 33.9 / 0.090 m = 377 N.
Solve for L:
L = Fx / [(2/Ãâ‚¬)IBz] => 377 /[(2/Ãâ‚¬) *44 * 0.5] = 377 / 14 = 26.9 m (which is equal to 2d)
Solve for length of wire per phase per tooth (Lpp):
Lpp = Lw / 3 / 7 => 26.9 / 3 / 7 = 1.28 m which is also equal to 2d,
1.28 / 2 = .6406 m / 640.6 mm as a single diameter.
With Perimeter determined by CAD; Calculate ID:
Pc = 75.5 mm = 2 Ãâ‚¬ r => r = Pc / (2 Ãâ‚¬) = 75.5 / (2 * Ãâ‚¬) = 12 mm
We want diameter; 2 * r = d => 2 * 12 = 24 mm
A crude estimate of the number of turns per tooth would be 640.6 / 24 = 26.7 Turns

Using 4:1 Flat Wire, 20 AWG = 0.39 High x 1.56 wide; the Archimedes Winding Calculator suggests that 26.7 turns with an ID of 24 mm = 2.92 m of conductor, and a 45.2 mm OD.

The first thing we notice is that OD of the winding is really tall, and what we want is closer to a square fill. There is plenty of room for a second winding. LetÃ¢â‚¬â„¢s round up the number of turns to the next even value divide by two; 28 / 2 = 14 turns x 2 rows. This comes out to 1.32 m of conductor with an OD of 35.3 which is much better, but we still have a ratio of 11:3 height-to-width which is nearly a factor of 4.

Throw in the Monkey wrench:
This has been all fun and good so far but the truth is I want this motor to handle a good blow, like the windstorm I experience on my Road Trip to California. I also donÃ¢â‚¬â„¢t want it to get hot when running WOT in the desert after I pass all them neckkÃ¢â‚¬â„¢id hippies out at Burning Man running away from the loooong arm of the law. So what if I took the hp rating and added another 50% to cover my worries; thatÃ¢â‚¬â„¢s a pretty good engineering margin of error. We throw that all into the mix and pretty soon this design begins so see some issues beyond the rows of turns.

So~ how do we make this critter more flexible? The answer isÃ¢â‚¬Â¦

... KF
(Don't you love a good mystery?)
Last edited by Kingfish on Fri Mar 04, 2011 2:59 pm, edited 1 time in total.
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The hands acquire shakes, the shakes become a warning.
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### Re: Doing the Math

Plan-D

Ã¢â‚¬Â¦We need to optimize this design; slow the wheel down and reduce the amount of current per cycle.

Revision:
32 Poles / 16:1, 30 Teeth / 10 teeth per phase
Repeating calcs we did for Plan-C:
Magnet length = 20 mm; Â½ length = 10 mm.
r = (200 / 2) Ã¢â‚¬â€œ 10 = 90 mm / 0.090 m.
F = Ãâ€ž / r = 33.9 / 0.090 m = 377 N
L = 26.9 m (which is equal to 2d).
Solve for length of wire per phase per tooth (Lpp):
Lpp = Lw / 3 / 7 => 26.9 / 3 / 10 = 0.897 m which is also equal to 2d,
0.897 / 2 = .4483 m / 448.3 mm as a single diameter.
With Perimeter determined by CAD; Calculate ID:
Pc = 64 mm = 2 Ãâ‚¬ r => r = Pc / (2 Ãâ‚¬) = 64 / (2 * Ãâ‚¬) = 10.2 mm
We want diameter; 2 * r = d => 2 * 10.2 = 20.4 mm diameter of the winding (dw).
The number of turns will be Lw / dw => 448.3 / 20.4 = 21.98 Turns => letÃ¢â‚¬â„¢s make it 22 Turns even.

Using 4:1 Flat Wire, 20 AWG = 0.39 High x 1.56 wide; the Archimedes Winding Calculator (AWC) suggests that 22 turns with an ID of 20.4 mm = 2.04 m of conductor, and a 38 mm OD. The actual thickness of the windings is calculated as
(OD Ã¢â‚¬â€œ ID) / 2 => (38 Ã¢â‚¬â€œ 24) / 2 = 8.8 mm

It turns out that a 8.8 mm thickness is too much for that many windings and it will not fit properly. Once again I suggest we take the number of turns and divide by two so that we can have two rows.

22 /2 = 11 turns. Calculate new OD and thickness: AWC suggests that 10 turns with an ID of 20.4 mm = 0.87 m of conductor, and a 29.4 mm OD. The actual thickness is
(OD Ã¢â‚¬â€œ ID) / 2 => (29.4 Ã¢â‚¬â€œ 20.4) / 2 = 4.5 mm
The 4.5 mm thickness fits perfectly. The total width of the windings is 1.56 * 2 = 3.12 mm. The thickness and the width are fractionally close to 1.5:1. It is better to have a 1:1 ratio, but this is pretty good as it is.

If we really needed to squeeze a little more out of this winding, we could reduce the ID by an offset of 1 mm, which yields a new perimeter of 57.7 mm, which gives us an ID of 18.4 mm. 448.3 / 18.4 = 24.4 turns; round up to next even number and divide by two yields 13 turns. The AWC suggests 13 turns takes 0.98 m of conductor, and a 29 mm OD. Actual thickness is (29 Ã¢â‚¬â€œ 18.4) / 2 = 5.3 mm, and that falls inside the perimeter of the original calculation by more than a whisker. LetÃ¢â‚¬â„¢s throw one more turn on it:

The AWC suggests 14 turns takes 1.07 m of conductor, and a 29.7 mm OD. Actual thickness is (29.7 Ã¢â‚¬â€œ 18.4) / 2 = 5.67 mm. When drawn up in CAD, the air gap between the windings is slightly more than 1/16-inch / 1.59 mm which by several sources is the minimum given for airflow. The amount of turns has improved from 11 to 14; 14/11 = 27% improvement which is indeed a nice margin. Furthermore the winding thickness does not exceed our preset 200 mm working diameter for the hub. Another item to consider is that we increased the gearing ratio from 10 to 16.

Tire rotation is 7 rps, therefore 7 * 16 p = 112 Hz
The Current also drops: Itooth = Itotal / 16 / Ã¢Ë†Å¡3 => 44 / 16 / Ã¢Ë†Å¡3 = 1.6 A.

The 20 AWG Flat Wire spec was left unchanged to help depress the internal resistance.

I like this design very much; weÃ¢â‚¬â„¢re reduced the magnet size, maximized the effective torque arm, reduced the Current per Phase per Winding, and we have reduced the z-distance between the rotors which increases our flux density to the point that I can safely use temperature-tolerant magnets.

Now about that money wrenchÃ¢â‚¬Â¦ If I want to add 50% more torque to this motor, how would I do it?
Are you still with me? KF
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The hands acquire shakes, the shakes become a warning.
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### Re: Doing the Math

I'm getting a little confused following the details of your calculations for the various coil dimensions. I think I'm also getting a little confused because your calculations still seem to assume the spiral-type winding, rather than the racetrack/sector shape that I was under the impression was the current choice.

Let me try running through some numbers myself to see if we can get on the same page.

For your plan D, the magnets are 20 mm long and you calculate that we need ~900 mm of wire per coil (per "tooth") for the desired torque.

I want to start from the opposite direction by figuring how much space we have. There are a total of 30 coils (10 per phase, 3 phases) around the stator. That works out to 12 degree sectors for each phase. At the inner radius of 80 mm that's an arc length of 16.6 mm and arc length of 20.8 mm at the outer radius of 100 mm. Combined with the magnet length of 20 mm that gives us an almost-square area in which to fit the desired coil.

Assume we are using the sector-shaped windings. We need 900/20/2 = 22.5 turns to get the desired torque (20mm per side, total of 40 mm per turn). Round up to 23. Using your wire dimensions of 0.39mm x 1.56mm, a stack of 23 pieces of wire is ~9 mm high. So, we can't fit the desired number of turns in a single layer. 12 pieces would be 4.7 mm high. That might be okay, and is ~3.1 mm wide. It's important for there to be an enclosed area in the center of the coil. We only get net torque if the two sides of the coil experience different flux, so if they are tight against each other not much will happen.

Note that the above coil dimensions don't leave any spacing between adjacent coils. Putting 30 coils within a 200 mm diameter seems pretty tight to me. I think reducing the number of coils, making the diameter larger, or using an overlapping winding would make it easier. An overlapping winding would not be too difficult. You can just make 2 stators each with half the number of coils, then stack them together offset by 1/2 coil and wire appropriately.
Eric

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### Re: Doing the Math

Eric, I knew I should have enclosed a picture. Figure 12 represents both the original and the modified Plan-D Windings. They are still a modified racetrack shape with the full-radius on the inside about as small as I dare; about 1.75 mm R. The skinny one on the right is the original with 11 turns, and the chunky one on the left has 14 turns. This is as tight as I can make it, though I donÃ¢â‚¬â„¢t know enough about winding physics to say if this will work or if it is optimum. I completely understand that some space is required.

OK, so you are correct about the second stator, and in fact you mentioned that as an option early on (forgive me: I was deliberately ignoring the suggestion so we could focus where directed). For the monkey wrench problem where I desire 50% more power to cover my bases Ã¢â‚¬â€œ the only way I could see getting this accomplished is with a second stator design. The direct benefit is that we can split the load so-to-speak between the two stators by connecting the phase coils in series.

In the hub design there is room for more than one stator. In fact Ã¢â‚¬â€œ I have room for four stators! See Figure 13.

In the beginning of the thread I believe I mentioned that I wanted to have the motor width limited to 50 mm max. LetÃ¢â‚¬â„¢s do the math:
Two rows per winding =>
Using 4:1 Flat Wire, 20 AWG = 0.39 High x 1.56 wide; 1.56 * 2 = 3.12 mm wide (theoretically)
If I use Â¼-inch /3.175 mm thick material, probably aluminum 70XX for high-strength/low-deflection, the turns will comfortably fit within the width of that stock material. We'll probably have to burnish off a tiny bit to match the winding.

• Air gap between the winding and the magnet surface is 1 mm, typical.
• The thickness/height of the magnets is 4 mm.
• The distance between the two opposing magnetic faces is 5.12 mm.
• The outer face, the thickness of the exterior rotor is 3/16-inch / 4.7625 mm thick Ã¢â‚¬â€œ at least where the magnets are located.
• The thickness of the material retaining the magnets between the stators is irrelevant because the forces are balanced; the magnets are slightly thicker than the retaining material. Because the magnets are short, the beam is about 1-inch / 25.4 mm and weÃ¢â‚¬â„¢re just not going to see a whole lot of deflection given the material and thickness.
• The total distance from exterior face to exterior face is 50.08 mm wide. I can live with the 0.08 mm.*

Thoughts...
I studied the multi-stator concept early on when I dabbled with the thought of building a motorcycle-class ebike knowing that I would need at least double the horsepower per motor of whatever I came up with for the ebike. The other factor we need to evaluate is that I am building a 2WD. The point is that there is a lot of flexibility with what we can do with an AF motor design. For my needs, I want the motor to be fast, robust, and not get terribly hot.

I am also sensitive to weight, and I came up with a slight modification where the magnets are only 3 mm high*, as well as calculating the Flux Density for a 6 mm air gap between the magnetic faces instead of 5 mm. All variations and permutations still predicted that the Flux Density would remain above the 0.5 Tesla values that we have been using throughout this thread. By reducing the gap, I have been able to tone-down my expectations and plan for a lower-Tesla magnet with higher thermal tolerance (120-140*C).

*BTW Ã¢â‚¬â€œ reducing the magnet height to 3 mm also reduces the width of the hub by 5 mm, therefore the new width could become 45.08 mm.

Does this make it clearer?
On the Cat-Bird Seat, KF

PS - I still have questions!
Last edited by Kingfish on Fri Mar 04, 2011 3:00 pm, edited 1 time in total.
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It is by the beans of Java that thoughts acquire speed.
The hands acquire shakes, the shakes become a warning.
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### Re: Doing the Math

I was never any good at juggling, and I think the same principle applies here. I'll focus on one point first until we're in agreement, then we can move on.

I think the winding shape is a good place to start, since the winding geometry defines so many of the other motor characteristics. I think we previously agreed on one basic fact: assuming the modified-racetrack winding shape, the effective length for torque production is 2x the magnet length. For the current 20 mm long magnets, we get 40 mm of useful length per turn in the windings.

Now, regarding the spacing/enclosed area part. Let's use a very simple example. Assume for a second a hairpin-shaped piece of wire, that is a very narrow U shape with long legs. Current flows up one leg and back down the other. Let us further assume that the U is immersed in a uniform and perpendicular magnetic field, such as that of the AF motor. The cross-product of the force equation tells us that the two sides of the U will produce a force in opposing directions. For the uniform magnetic field case, there is no net force. If it were somehow possible to create a +z-directed field on one leg of the U and a -z-directed field on the other leg, that gives us the maximum force case. Reality will be somewhere in between, but we obviously want to be closer to the second example than the first.

I don't know if there is a common design rule-of-thumb or other guidance for the winding shape in this context. Your AF text may have some guidance here. But, it seems to me that based on this common-sense analogy, you would want the spacing between the two sides of the winding to be something like the spacing between adjacent magnetic poles (this would be 2N in your Halbach array). If they're much closer than that I think torque production will start to suffer. Put another way, this is probably the reason most motors usually have a similar number of teeth and magnetic poles. I think if you want a significantly higher number of windings/teeth, then you have to go to a distributed/overlapping design. Note that this is distinct from and not equivalent to a two-stator design. We can discuss that separately.
Eric

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### Re: Doing the Math

Good points

Windings Re-Work:
I am trying to avoid the overlap windings because the complexities multiple over the non-overlap, however I understand that the leverage would be greater because the legs of the windings are farther apart. There needs to be at least 1* of arc between the windings to promote air flow and cooling. Ideally the width and height cross-section should be square, actually round, but round is difficult to control.

Certainly we could get more area to work with if the bounding Radius were allowed to expand, however I think that the 2806 hubs are already ungainly in diameter. The only alternative is to move the magnets to the rim Ã¢â‚¬â€œ and maybe that should be explored a bit later Ã¢â‚¬â€œ though I suspect that is another can oÃ¢â‚¬â„¢worms.

Another scenario is to reduce the size of the conductor yet again. If we dropped down from 20 AWG Flat Wire to 24 AWG, the width goes to 1 mm which would allow 3 rows. LetÃ¢â‚¬â„¢s calculate the results and see where the dust settles.

Whether we pick 22 or 23 turns, going to 24 turns is easily dividable by 3, therefore letÃ¢â‚¬â„¢s use 8 turns per row with a conductor height of 0.25 mm. I am going to calculate using an ID that accepts 24 mm for each leg which to be longer than the height of the magnet; it only affects the length of the conductor which we need for mass and resistance calculations. WeÃ¢â‚¬â„¢ll need to iterate until we can get the maximum ID for the greatest distanceÃ¢â‚¬Â¦ LetÃ¢â‚¬â„¢s use the initial 20.4 mm ID. We want to end up with a perimeter of about 93 mm, or about 30 mm OD.

The AWC suggests that with 0.57 m of conductor we can achieve 8 turns with an OD of 24.6 mm. OK Ã¢â‚¬â€œ this is a good start, so letÃ¢â‚¬â„¢s move the ID out and try againÃ¢â‚¬Â¦

Saving a bit of time, I massaged the values until I came up with:
ID = 25.7 mm, conductor length = 0.71 m, OD = 29.99 (or 30) mm, and turns = 8.08. The cross-section is (30 Ã¢â‚¬â€œ 25.7) / 2 = 2.15 mm x 1 mm x 3 rows = 2.15 mm high x 3 mm wide. Not optimum but doable.
If you will oblige my fancy, I would like to calculate with an ID of 24 mm which when matched with an OD of 30, we would get the 3 x 3 cross-section; how many turns will that provide?

AWC suggests that with an ID = 24 mm, the resultant OD is given as 29.996 mm, using 0.979 m of conductor and providing 11.5 turns. This means that our start and end turns are on opposite sides. Allow me to reduce that to 11 turns even: ID = 24.25 mm and the conductor length becomes 0.945 m.

Unfortunately the current carrying capacity of 24 AWG is about 1/3 of 20 AWG, and yet Ã¢â‚¬â€œ the value is more than twice what we expect weÃ¢â‚¬â„¢ll need for this design.

Caveats:
Originally the design was for one row per winding, however now we are considering 3 rows. I have a concern about resistance and inductance and would like very much to review those values, and how they could affect the final plans.

Eric, can you live with a layout as shown in Figure 14 where the cross-section is close to ideal, the separation between the windings is enough for airflow, and the distance between the up and down legs are at the maximum? The overall turns advances from the original 22 or 23, to 33 turns by using smaller gauge flat wire. If weÃ¢â‚¬â„¢re good on this I think it would be prudent to calculate resistance, inductance, and changes to Current and voltage.

Best, KF
Last edited by Kingfish on Fri Mar 04, 2011 3:00 pm, edited 1 time in total.
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### Re: Doing the Math

Kingfish wrote:Eric, can you live with a layout as shown in Figure 14 where the cross-section is close to ideal, the separation between the windings is enough for airflow, and the distance between the up and down legs are at the maximum? The overall turns advances from the original 22 or 23, to 33 turns by using smaller gauge flat wire. If weÃ¢â‚¬â„¢re good on this I think it would be prudent to calculate resistance, inductance, and changes to Current and voltage.

The magnet sectors drawn in red in your FIg 14 - what do they represent? I mean, does each sector represent a "full pole" of the Halbach array (a L-S or R-N combination), or is each sector just a single magnet? The spacing looks like it should be good if those are poles.

The triple layer of 24 AWG wire looks like it would work out well. Depending on what Kv you want to achieve, you could connect the layers in parallel and get approximately the same current capacity of the single 20 AWG, but with 50% more turns.

Kingfish wrote:Certainly we could get more area to work with if the bounding Radius were allowed to expand, however I think that the 2806 hubs are already ungainly in diameter. The only alternative is to move the magnets to the rim Ã¢â‚¬â€œ and maybe that should be explored a bit later Ã¢â‚¬â€œ though I suspect that is another can oÃ¢â‚¬â„¢worms.

Just for kicks, I think that would be an interesting design exercise. You can imagine the opposite of a hub motor - sort of a "rim" motor. Construction would be tricky, but you can imagine a way where the magnets are integrated into the rim of the wheel so you get the maximum torque arm possible. If you could get past the construction details, that would be an excellent way to build a high-torque low-RPM motor with a very high pole count.
Eric

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### Re: Doing the Math

rhiteeo5 said
You can imagine the opposite of a hub motor - sort of a "rim" motor. If you could get past the construction details, that would be an excellent way to build a high-torque low-RPM motor with a very high pole count.

Arlo1 had the same idea awhile back....
http://endless-sphere.com/forums/viewto ... f=2&t=9966

no need for tourque arms if the stator is mounted to the chain stays
get some......

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

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### Re: Doing the Math

The magnet sectors drawn in red in your Fig 14 - what do they represent?

Eric, you have an attentive eye sir! TÃ¢â‚¬â„¢was caused from a bout of laziness and drew only the unified pole (2n) for simplicity: Represents single poles.

The triple layer of 24 AWG wire looks like it would work out well. Depending on what Kv you want to achieve, you could connect the layers in parallel and get approximately the same current capacity of the single 20 AWG, but with 50% more turns.

I think that makes for a great talking point that strangely circles back to a problem that I had right near the beginning of the thread Ã¢â‚¬â€œ and it's an issue that we have to address before continuing:

The Inductance (L) calculation for a single row spiral-winding aka a flat Archimedes spiral one-wire deep is given as:

L (ÃŽÂ¼H) = A^2 * n^2 / (8 * A + 11 * w)

Where
L = Inductance, measured in micro-HenryÃ¢â‚¬â„¢s (ÃŽÂ¼H)
A = Average radius of the coil*, m
Given as
A = [((OD Ã¢â‚¬â€œ ID)/2) + ID] /2
Where
OD = Outside Diameter
ID = Inside Diameter
n = number of turns
w = diameter of the wire

Now however we have a multi-row winding Ã¢â‚¬â€œ the intention being three sets of distinct turns rather than one wound coil which uses a slightly different formula to determine Inductance (L).

• I agree that deciding on Series or Parallel will make a performance difference. The factors that I struggle with however are that parallel inductors could suffer from eddy currents at higher frequency:
• Do we need to worry about this, and if so when does the frequency begin to assert negatively?
• The second issue is that I had a problem determining the correct value for Inductance (L) given the Wheeler formula.
• Third,
• of what use is L to us in the balance or design of the circuit, and how does this affect our ControllerÃ¢â‚¬â„¢s FET stage?
• Should it be the providence of the Motor to be compatible with traditional controllers, or should the FET stage accommodate a wide assortment of motors?
• It is a question of ownership: Which unit is responsible for compatibility?
Rim Motor:
OK, letÃ¢â‚¬â„¢s give that a serious review then, however IÃ¢â‚¬â„¢d like to save it as the desert with the cherry on top after we sort out the remaining design issues.

Thud: Chain stays vs. Torque Arm; Yeah that makes sense. The whole interface would require review, and I think that a non-wire-spoke rim would be best off the top of my little solar-powered propeller-beanie cap. We must save this discussion though because it is definitely worthy of exploration.

For now though, we have to finish the Math.
Best, KF
Last edited by Kingfish on Fri Mar 04, 2011 3:01 pm, edited 1 time in total.
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### Re: Doing the Math

Calculating the inductance of the triple-layer coil gets a little messy. You have to consider both the self- and mutual-inductance.

The self-inductance of each coil should be close to the value for the simple Archimedes spiral. The geometry isn't exactly circular, but it should give you a ballpark number. The mutual inductance in this case will be pretty high since the coils are identical in shape and tightly stacked. It won't be quite this high, but for the sake of argument we can assume that M=L, that is the mutual inductance is equal to the self-inductance. The polarities are the same so everything adds.

In the series case, the total inductance including the mutual terms will probably be ~8-9 L, where L is the inductance of the single coil. There are a total of 9 self- and mutual-inductance terms which are each close to L, so the net total will be somewhat smaller than 9L. For the parallel case, it looks like the total inductance would be similar to that of a single coil, again probably a little smaller. Mutual inductors are tricky but I think I got that right after some Wikipedia consultation.

Since L/R is the more important measure, we should consider that. The series case would give us 9L / 3R, so the total ratio is about 3x the single-coil L/R ratio. In the parallel case, we get L / (1/3 R), which is also 3x the single-coil L/R ratio. Realistically, since M will never be quite as high as L, both cases would probably have L/R somewhat less than the single-coil case.

Not sure what you mean regarding the parallel inductors and eddy currents. Can you elaborate?

I think in most cases the value of L is simply a consequence of the motor design. The various choices of material, geometry, etc. are driven more by specifications like desired torque and Kv, so L is simply "whatever it ends up being." Design being a series of trade-offs, if a particular L value is desired (or more likely the L/R ratio), compromises would likely have to be made in other characteristics. It's probably a more effective trade-off to just add external inductance if it's needed.

Being a coreless motor design, I think it's reasonable to assume that the L/R ratio (the time constant) will be fairly low. That would make this motor tend to be more demanding of controllers and require a higher PWM frequency for effective control. It might be well within the range of what a common controller can handle or might not. Hard to say at this point!
Eric

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### Re: Doing the Math

FWIW - I performed a quick mass calculation of the current design as a single- and dual-stator motor on this thread.

Aluminum parts L-R: Internal Rotor, Rotor Cover, Stator

The middle item is the cover for the hub and acts as the primary mounting plate for the magnets.
Note: All designs are preliminary and not to scale (NTS).

Back in a flash... KF
Last edited by Kingfish on Fri Mar 04, 2011 3:02 pm, edited 1 time in total.
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### Re: Doing the Math

rhitee05 wrote:<snip>
Not sure what you mean regarding the parallel inductors and eddy currents. Can you elaborate?
<snip>

Yes Axial Flux Permanent Magnet Brushless Machines Page 170; 5.8 Eddy Current Losses in the Stator Windings
(I own the book, though this is a quick online reference)

On P 172, section 5.8.2 they discuss minimizing eddy losses with parallel wires, Litz wire, or flat wire.

Then it goes on about Reduction of Circulating Current Losses in 5.8.3, and using tightly turned wires, citing a motor that spins at 400 rpm (hint hint), and goes on about relative issues with parallel windings.

Summation: There is no silver bullet. Litz wire has its' issues with poor filling, and I am struggling to find it in the 3 mm widths; it may not be possible as an option. IMM - Flat Wire in series is the leading candidate.

Thank you for explaining Inductance/Reactance/Mutual Inductance. If an inline inductance winding is required, would that item be placed on the FET board or within the motor?

Given the potential issues with Eddy & Circulating Current, should we proceed as all windings are in series and figure out resistance, voltage, and current?

Best, KF

PS - my tail is starting to wag again...
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### Re: Doing the Math

FEMM Studies

I took a bit of a holiday today and spent the afternoon teaching myself how to wiggle about in FEMM 4.2:
• Created a side profile of a 3-Rotor AF motor and exported out in DXF format.
• Imported into FEMM
• Spent many hours scratching my pointy head and rewinding the twirly propeller on the beanie cap experimenting with different aspects of the application. (Someone could make a mint rewriting the User Manuals.)
Anyway Ã¢â‚¬â€œ I finally ciphered most of it out. I had a question about the Materials Library: ThereÃ¢â‚¬â„¢s a profile for NdFeB 52 and another for 40, but I had been planning on using 45 or 48. In the end I decided to model both 40 and 52 and take the average between the values.

Allow me to direct your attention to the figures belowÃ¢â‚¬Â¦

Figure 16A employs the NdFeB 52 magnets; in this diagram the Flux Density is highest between the magnetic faces Ã¢â‚¬â€œ and estimates to be 0.65 to 0.79 T which is very good.

Figure 16B is the same as 16A except that there are only 2 rotors so we could compare the two configurations; in this diagram the Flux Density has a wider variation in field strength Ã¢â‚¬â€œ and estimates to be 0.55 to 0.8 T which is good.

Figure 16C employs the NdFeB 40 magnets; in this diagram the Flux Density is strong between the magnetic faces Ã¢â‚¬â€œ and estimates to be 0.54 to 0.66 T which is good.

Conclusions:
• Adding internal rotors stabilizes & reinforces the field strength, providing more consistent distribution. I am tempted to model 4 and 5 rotors now just to see the affects.
• There wasnÃ¢â‚¬â„¢t a monstrous difference between 40 and 52 magnet strengths; we can deduce that a 3-rotor motor will have a Flux Density of 0.6 to 0.7 which is admirably better than the 0.5 we have been batting around in our calculations.
• Aluminum, Copper, and Air have little effect on the Flux path Ã¢â‚¬â€œ which is very good.

Notes:
• I replaced the Aluminum outer rotor material with Magnetic Stainless 455, and the flux Density spread itself out more evenly across the divide. However the field strength was lowered by 0.05 which confirms what the authors have stated in the book that IÃ¢â‚¬â„¢ve been using. On the flip-side there was zero-flux leakage external to the system.
• It would be interesting to study how the width of the inner magnets affects the field strength. I would presume that narrowing the magnet would have a similar effect as inserting a very high powered magnet, such as the differences observed between the 40 and 52 flux lines: The stronger magnets had better confinement.
• I think it would also be of value to recalculate the Current required given that FEMM has indicated we have a potential 20-40% improvement in field strength Ã¢â‚¬â€œ if for nothing more than academia.

Thoughts? KF
Last edited by Kingfish on Fri Mar 04, 2011 3:03 pm, edited 1 time in total.
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### Re: Doing the Math

FEMM is a fun tool, isn't it?

If you set up the model properties right, FEMM will calculate the force for you. For each of your coils you can add material properties for copper with such-and-such number of turns, such-and-such current.

The next step up is you can set up the model to allow you to move the rotors relative to the stator and plot the resulting data. I believe it's possible to generate a BEMF curve, for example. I'm a little fuzzy on the details but I think there is an example (either included with FEMM or available on their website) which goes through this for a radial-flux motor.
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### Re: Doing the Math

rhitee05 wrote:FEMM is a fun tool, isn't it?

If you set up the model properties right, FEMM will calculate the force for you. For each of your coils you can add material properties for copper with such-and-such number of turns, such-and-such current.

The next step up is you can set up the model to allow you to move the rotors relative to the stator and plot the resulting data. I believe it's possible to generate a BEMF curve, for example. I'm a little fuzzy on the details but I think there is an example (either included with FEMM or available on their website) which goes through this for a radial-flux motor.

Total fun! IÃ¢â‚¬â„¢ve been at this all morning and now afternoon. Getting faster at setting up the models tooÃ¢â‚¬Â¦

Eric, I saw that the example directory had a model in their like what youÃ¢â‚¬â„¢re thinking. I hope to get to that level shortly Ã¢â‚¬â€œ it would be a sure-fire way of validating the design before cutting metal wouldnÃ¢â‚¬â„¢t it?

FEMM Studies Ã¢â‚¬â€œ Part II
Being a busy little body, I spent a bit more time thinking about the FEMM output and decided to test out a couple of ideas.

Notes:
1. All exterior rotor magnets were upgraded to 4 mm tall.
2. I double-checked to ensure the air gaps between all magnet faces was 5 mm.
3. Unless otherwise specified all internal rotor magnets were 3 mm tall.
4. The materials of the stators didnÃ¢â‚¬â„¢t affect the output hugely therefore I removed them from the study for observation and clarity.
5. All magnets were modeled as NdFeB 52.
6. With all FEMM Analysis output, the Flux Density (FD) Lower Bound was set to 0.3, and the Upper Bound was set to 0.8; this cuts off the low and the high and better enhances the subtle density differences within the stator region. Warmer-Pinker colors = greater density.
7. Variations between the top and bottom Poles are likely caused by the linear layout; each model actually has 4 complete poles, though only 2 are shown.

Figure 17A: 3-Rotor/2-Stator. Resetting the exterior magnets to 4 mm definitely increased the Flux Density (FD) to about 0.65 Ã¢â‚¬â€œ 0.675 T.

Figure 17B: 4-Rotor/3-Stator. Adding another Rotor/Stator reduces the FD slightly to 0.60 Ã¢â‚¬â€œ 0.65 T. Also asymmetric anomalies begin to encroach Ã¢â‚¬â€œ however this could be an artifact produced by the incomplete model.

Figure 17C: 5-Rotor/4-Stator. This is the maximum number of Rotor/Stator pairing for this design. It is interesting to note that the FD continues to weaken, especially in the center. Perhaps this could be improved by placing a 4 mm tall magnet in the center rotor would re-boost the FD from the observed 0.60 Ã¢â‚¬â€œ 0.65 T.

Figure 17D: 4-Rotor/3-Stator. The reason for using 3 mm was to reduce the mass and the width of the motor. With the 4R/3S configuration I thought it would be fun to model 4 mm tall magnets all around. The most obvious result is more uniform and constrained Flux with higher concentrated peak density approaching 0.7 T.

More diagrams in a momentÃ¢â‚¬Â¦ KF
Last edited by Kingfish on Fri Mar 04, 2011 3:05 pm, edited 1 time in total.
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### Re: Doing the Math

FEMM Studies Ã¢â‚¬â€œ Part III

Figure 17E: Double-Halbach/2-Stator. Curious, I decided to explore a double-Halbach model. It was presumed the middle stator region would be nearly barren of flux, however peak FD is > 0.8 T.

Figure 17F: Double-Halbach/2-Stator Version 2. I removed the unused stator region for evaluation. The FD is again enhanced; if I changed the Upper Bound to 0.9 the peak density would still be higher though not quite 1.0 T. This is as good as it gets, unlessÃ¢â‚¬Â¦

Figure 17G: 3-Halbach/3-Stator. Plop in another Halbach and this is as strong as FD will get within the design space. This is some pretty monstrous peak density at the cost of a lot of magnets; the peak range is 0.8 T Ã¢â‚¬â€œ 0.95 T, with the center rotor getting the most benefit.[

Figure 17H: 3-Halbach/3-Stator. Exact same configuration as above except I changed the Upper and Lower Bounds to 0.4 Ã¢â‚¬â€œ 1.0, meaning everything in cyan is lower than 0.4 T and everything that is bright pink is => 1 T. Look at where the center stator would be! Now if we could just afford to build it. I would estimate this motor to output 6X the original design of a single Halbach/single stator.

Conclusions:
• Certainly increasing the magnet high improved the FD all around.
• Adding more internal rotors and stators is an efficient way of increasing capability however the benefit is not linear.
• Doubling and tripling Halbach pairs nearly doubles the FD but at the additional cost and weight penalty of 1.5X over a simple internal rotor.
Which one of these figures shall I go with to prototype?
Guesses? KF
Last edited by Kingfish on Fri Mar 04, 2011 3:07 pm, edited 1 time in total.
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### Re: Doing the Math

FEMM Studies Ã¢â‚¬â€œ Part IV

The more I thought about the magnet height the more I wanted to do a side-by-side comparison using the exact same bounding conditions for heights 3, 4, 5, and 6 mm.
• Therefore the Flux Density (FD) in the following figures are constrained between 0.5 and 1.0 T; if it is Cyan then the FD is < 0.5, and conversely if it is hot pink then the FD is >= 1.0.
• The air gap between the magnet faces is precisely 5 mm in all figures.
• The material used for Magnet modeling is NdFeB 52, therefore the calculations are optimistic. Through previous observations Ã¢â‚¬â€œ any figure given should be downgraded by 0.05 T to match NeFeB 45 Ã¢â‚¬â€œ 48 materials.

Figure 18A: Magnet Height = 3 mm. Generally speaking we can deduce that the FD is between 0.5 and 0.6 T. This height minimally meets our design considerations.

Figure 18B: Magnet Height = 4 mm. Good improvement in FD with better field containment and nice peak development approaching 0.75 T.

Figure 18C: Magnet Height = 5 mm. Very good FD with slight widening of the affected area, and with a definitive peak clearly above 0.8 T - even with the 3-Rotor/2-Stator layout.

Figure 18D: Magnet Height = 6 mm. Excellent FD with a broad affected area across the entire magnet width, and with a pronounced peak above 0.9 T with the 3-Rotor/2-Stator layout, and greater than 1 T with the single-Stator.

Conclusions:
• The progression of FD strength is nearly linear with material thickness.
• If we required only 2 hp output then the layout given in Figure 18A single-Stator would achieve that goal.
• If we desired to increase that to 50% more, being equal to 3 hp, the least expensive solution would be an upgrade to 5 mm high magnets has shown in Figure 18C left-side.
• Doubling the original design rating from 2 to 4 hp could employ the right-side of Figure 18A for the least weight, though we might as well take the performance improvement weÃ¢â‚¬â„¢d achieve with 4 mm high magnets in Figure 18B right-side; essentially (1.5) Â² or 2.25X = 4.5 hp. Both are cost-effective solutions: Doubling output with only 25% more magnetic material. Admittedly a double-stator is more complex than a single. Simple doubling could be achieved by doubling the height as indication with Figure 18D left-side.
It is interesting to note that the right-side of Figure 18D could in theory produce (1.8 ) Â² or 3.24X more power (~6.5 hp) with the addition of 1.5X more magnetic material. Far more than what we need for a typical bicycle, yes?

More in a bitÃ¢â‚¬Â¦ KF
Last edited by Kingfish on Fri Mar 04, 2011 3:09 pm, edited 1 time in total.
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### Re: Doing the Math

FEMM Studies Ã¢â‚¬â€œ Part V

Limits to Height:
It is tempting to continue study the effects of magnet thickness beyond 6 mm, however as previously indicated there becomes a point of diminishing returns where further increase does not equate to equal payback. The limit of maximum improvement is around 8 mm high, with additions become infinitesimally smaller thereafter. The other problem encountered is that pull-force of the magnets becomes a danger to manufacturing during assembly or repair.

Hmmm, now IÃ¢â‚¬â„¢m curiousÃ¢â‚¬Â¦

OK, in Figure 19 the diagram is rendered with the boundaries 0.0 to 1.2 T, meaning that we have peak FD higher than 1.2 T within the Stator region for either a single- or 2-stator solution. However is another problem with this layout and thickness best exemplified in the 2-stator on the right-side layout: The FD bleeding over in the region between the poles is significant. Could this adversely affect performance

Ultimately what we really want is a light-weight solution for bicycles, and then motorbikes. With that in mind I will likely chose between 4 or 5 mm high magnets.

In review, weÃ¢â‚¬â„¢ve explored Material Quality (Tesla-rating), Radius, Length, and Height of magnets. What happens if we change the Width?

Curiously KF
Last edited by Kingfish on Fri Mar 04, 2011 3:09 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: 3865
Joined: Wed Feb 03, 2010 11:23 am
Location: Redmond, WA, USA, Earth, Sol, Orion–Cygnus Arm, Milky Way. Age: > yesterday < tomorrow

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