Doing the Math

Electric Motors and Controllers

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.
Eric

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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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The hands acquire shakes, the shakes become a warning.
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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.
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Re: Doing the Math

FEMM Studies Ã¢â‚¬â€œ Part V

Limits to Width:
I would like to explore one other dimension of the static geometry with FEMM before continuing onward: What happens to the FD if we narrow the L-R magnets of the Halbach array

With the Figures 16 through 19 the Width of the magnets was calculated by taking the measure at the midpoint of length at the approximate radius of 75 mm, and equals about 14 mm.

LetÃ¢â‚¬â„¢s study what happens when we reduce this value by iterations of 2 mm for a 4 mm high single- and 2-stator solution for the L-R magnets. Note that the value of N-S magnets will conversely increase by 2 mm.

Figure 20A: There are no changes; equidistance Widths, nice peak, looks good.

Figure 20B: N-S poles are 2 mm wider, as the L-R poles are 2 mm narrower. The left-side is already beginning to see artifacts and separation of the peak. This does not bode well.

Figure 20C: N-S poles are 4 mm wider, as the L-R poles are 4 mm narrower. The left-side has pronounced separation of the peak, and the right-side is beginning to show a similar pattern.

Conclusions:
Changing the Width to other than equal lengths is a not a good idea. However, short of modeling we could only guess without rigorous math; FEMM makes it easy to quickly grasp the effects.

Oh well, so much for that idea. Gosh, think of the money I just \$aved myself.
Moving right alongÃ¢â‚¬Â¦ KF
Last edited by Kingfish on Fri Mar 04, 2011 3:10 pm, edited 1 time in total.
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Re: Doing the Math

FEMM Studies Ã¢â‚¬â€œ Part VI

Manufacturing:
This morning I began to consider manufacturing issues, specifically Ã¢â‚¬â€œ what if the pull-force of the magnets within the center internal rotor becomes stronger than the adhesive. In my mind I presume that the forces will be equal and balanced upon the center rotor:

Left-side magnets pulls equally towards the center magnets which in turn pulls equally towards the right-side magnets when the forces are in the same direction, and the reverse would happen a few degrees previous and hence around the entire circumference.
Unfortunately we have only the thickness of the center rotor material and matching magnet to hold the units in place. Therefore I began to explore the center rotor as having two identical thinner magnets on each side with the entire face as an attachment surface.

What then would be the consequence of geometry changes upon the Flux Density (FD)?
In this next series of diagrams I set the minimum boundary to 0.5 T because I only wanted to concern myself with minimal required FD = 0.5, however as the studies progressed I needed to bump the higher level up from 0.8 to 0.9 Ã¢â‚¬â€œ as will be evident momentarily. (Imagine I have been modeling this stuff since 9 AM this morning, and it has me geeked-out big time ).

Figure 21A: 4 mm high magnets. The Left-side has been modeled previously as Figure 18B right-side, and IÃ¢â‚¬â„¢m doing so again for a baseline. Right-side divides the center magnet in half as two 2 mm magnets separated by a 3 mm thick internal rotor, and we can easily observe field weakening.

Figure 21B: Same 4 mm magnets on left and right, except that left has 3 mm high pairs and right has 4 mm high pairs. With this diagram we can observe definite improvement in FD proportional to the thickness. However I saw another possible pattern emerge and I wanted to explore a new path:

Can you see the X-path forming between the center poles? Likewise it appears that the FD within the stator area is thinning in a peculiar manner. What if we narrowed the width of the center poles to help focus the peak?

Figure 21C: Same as Figure 21B except we have narrowed the center poles evenly by 2 mm. The FD is better constrained, but observe the effect between the center poles! What happens if we squeeze this down a bit more?

Figure 21D: Same as 21B except now weÃ¢â‚¬â„¢ve reduced the center poles by 4 mm evenly. I was curious about the angular tack of the FD and thought it might be an anomaly of the model and therefore completed the magnetic circuit with imaginary end-poles Ã¢â‚¬â€œ however it had zero impact upon the model. The big difference between the previous figure and this is that the trend of constraint continues with the shrinking of the width.

Figure 21E: Taking this one step farther I decided to thicken the N-S poles of the Halbach array by 1 mm (5 mm total) and observe the effects, mainly out of the concern that the FD was becoming less perpendicular to the magnet face. However we can abstract further constraint in what appears to be lensing or focusing of the FD.

Most intriguing Ã¢â‚¬â€œ I skipped lunch to continue modelingÃ¢â‚¬Â¦
~KF
Last edited by Kingfish on Fri Mar 04, 2011 3:12 pm, edited 1 time in total.
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Re: Doing the Math

You've probably reached the point in your modeling where its useful to actually generate some (numerical) metrics for comparison and optimization.

Obviously the axial flux density is an important metric. Instead of just looking at the pretty colors you can have FEMM display numerical values along an arbitrary contour. What you could do is to add a couple of nodes to your model which define a cut through the center of the stator. In the post-processor, you can then define this contour. You can have FEMM generate a plot of either the total magnitude of B or the normal component of B (more useful) along this contour. You can also have FEMM calculate a line integral of B dot n over the contour.

To get a little more advanced, if you define the coil properties, current, and so forth FEMM will calculate the force for you as well. There are a lot of things you can do with the post-processor to get more useful results than just the pretty pictures.
Eric

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rhitee05
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Re: Doing the Math

Hi Eric
Yeah I know IÃ¢â‚¬â„¢ve gone plot-crazy here, and I have more to post. Personally I donÃ¢â‚¬â„¢t care much for Cyan and Magenta but damned if I know how to change the palette without Hexplorer. It wouldn't take much to sex-up the output

I have been reading the docs and pouring over what you suggest. I think I need a bit of hand-holding:
To get useful information from the windings, other than craft the physical representation, how do I indicate current and direction? Do you have an example I can borrow/learn from? Or is it simply setting material and properties?
Careful: Once you teach me how to fish I might go whale-hunting!
Thanks a bunch, KF
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Re: Doing the Math

Why don't you pick one of your models as the example and we can use that to demonstrate on. Post it here if you like or PM me and I'll give you my email.
Eric

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Re: Doing the Math

Awesome!
Thanks Justin at http://www.ebikes.ca/
Thanks to Methy at http://www.methtek.com/
And Dave who has Nomex, fets and current sensors etc. STUFF
My YSR build. viewtopic.php?f=10&t=18183
RC lipo and most other types of Lithium batteries you MUST know your individual cell voltages charging and discharging.
Don't keep them under your bed or were you cant afford smoke or fire!
Never above 4.2v never below 2.7v EVER!!!

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Re: Doing the Math

FEMM Studies Ã¢â‚¬â€œ Part VII

Eric, I wish to post this series as the last of the physical studies before moving on to magnetic circuits.

Lensing Effect:
In the next series, the boundaries remained the same: 0.5 to 0.9 T. I observed that the geometry lent itself to lensing when in a particular configuration. Many years ago I became an early student of magnetic lensing when working directly across the street from the worldÃ¢â‚¬â„¢s largest magnetic fusion facility at Lawrence Livermore National Laboratories in 1985. The question in my mind is how malleable is the field shape and strength given the primitive shapes at our (theoretical) disposal?

Figure 22A: Same as 21D with 4 mm high Halbach magnets, 3 mm high center magnets on the left, and 4 mm high center magnets on the right. In addition, I have added thin Focusing Magnets (FM) to each external rotor; left-side is 1 mm high, and the right-side is 2 mm. This does a nice job of aligning the FD perpendicular to the magnetic face, and homogenizes the distribution within the stator region, with the right-side profoundly enhanced. But this arraignment is very complicated, and IÃ¢â‚¬â„¢d wager expensive. Nice peak density though!

Figure 22B: LetÃ¢â‚¬â„¢s step back and evaluate the focusing magnets with a single center magnet array. Left-side is 4 mm high with a pair of 2 mm high FM. Right-side uses 5 mm high instead of 4 mm. Check out the peak FD! When I modeled this arrangement I had to go reset the boundaries for Figures 21 and 22. Notice the profound X-Pattern between the Halbach arrays. Maybe the center magnet width could be reduced; letÃ¢â‚¬â„¢s investigate.

Figure 22C: Oh this is definitely a step-backward and very much in the same vain Figure 20 series. Less is not more. What if we reverse and grow wider?

Figure 22D: On the left 4 mm high magnets, Halbach and center poles are the same width. The FM are unchanged at 2 mm high. Right-side is 5 mm high. The left-side had a broader density though I think Figure 21B has better peak, however the right-side is arguably enhanced. LetÃ¢â‚¬â„¢s take this one more step.

Figure 22E: Left-side 4 mm high Halbach and center poles, with 2 mm high FM on each side and again as a pair applied to the center rotor. On the right-side I increased the center pole to 6 mm high. Both sides experience peak FD near 0.9 T.

Conclusions:
The FD can be focused more effectively by placing smaller geometrics in key locations. However it is unfortunate that with manufacturing being what it is that we are limited with using rectangular shapes as oppose to curves or splines - and therefore would like to conclude this aspect of investigation.

Leaving this subject; moving on to Magnetic Circuits.
~KF
Last edited by Kingfish on Fri Mar 04, 2011 3:15 pm, edited 1 time in total.
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Re: Doing the Math

FEMM Studies Ã¢â‚¬â€œ Part VIII

Magnetic Circuits:
After studying the examples in FEMM, I selected two configurations and reinstated the stator coil geometries.

Eric, allow me to explain what I did to create these models.
1. We are focused on 2-Stators/3-Rotors;
2. Stators follow the Plan-D except we now have double the windings, therefore the Current drops from 1.6 A for a single Stator to 0.8 A for a double. Additionally, I defined a single circuit of i = 0.8 A
3. All magnets are NdFeB 52, in either 4 or 5 mm thickness.
4. The Winding material is 1 mm Magnetic Wire with an Electrical Conductivity of 58 MS/m.
5. I am using a new material definition of Aluminum 7075 with an Electrical Conductivity of 20.88 MS/m.
6. Each Copper section (there are two-pairs) belongs to Circuit i with 33 turns.
7. Plots are 0.5 to 0.9 T

Figure 23A: 4mm high magnets. Looks great! We can observe what might be the influence of the winding upon the Flux Density.

Figure 23B: 5 mm high magnets. This has an anomaly and appears unaffected by the winding. I rebuilt the model from scratch and there was no change.

Eric, I am going to PM the files over to you; maybe you can spot my error. I did do a little bit of toying around but I will wait for your direction because I feel like IÃ¢â‚¬â„¢m fly-fishing in the dark without a flashlight.

Thanks, KF
Last edited by Kingfish on Fri Mar 04, 2011 3:16 pm, edited 1 time in total.
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Re: Doing the Math

There was only one thing wrong with your model - direction. You modeled the coil correctly, but on one side the number of turns needs to be negative to indicate the opposite current direction. Even so, you wouldn't see any difference. Try looking at this:

Stator Only.png
Stator Only - all magnets replaced by Al 7075

If you look at the scales, you'll see that the fields due to the stator currents are a very small fraction of the PM fields. The change in the combined fields is almost a rounding error. Now, for some analysis using the model you sent!
(AF-Mag-2S-4mm-Circuit)

The first useful result is to simply show what the normal component of B is at the stator. I added a couple of nodes at either side of the magnet arrays, midpoint between the first and second rotors where the first stator would be located. On the analysis screen you can use the contour tool to define a contour along any path between nodes (as many nodes as you like). Direction matters (a sign change), I went bottom-to-top. You can then hit the plot button, which gives you a selection of values to plot along the length of your contour. If you select B.n, it gives the plot shown below:

Stator Contour.png
Normal component of B along stator

Just as the graphics show, there is a nice even distribution of flux across each pole.

Another useful thing would be to see how much flux is passing through our coil. Using another set of nodes we define a contour through the center of the coil, and use the line integral function. If we choose B.n it gives us the total flux in Wb and the average flux in T:

Coil Flux.png
Flux coupled through coil

Now we're getting somewhere more useful. We can make one more calculation which gets us the number we're really looking for - torque. Using the block select function on the analysis page, we can select the 4 blocks which make up the coils. Clicking the integral button gives us a number of options. We can choose Lorentz Force, which does the J x B integral for us:

Lorentz Force.png
Calculation of force on coil

We can use this number to get an estimate of torque production. But, hang on one second before you hit the calculator.

It's useful to note here that only the upper 2 coil blocks are really contributing anything to the force. You can verify this by re-calculating the integral with only those blocks selected, but it should be obvious since the flux density is very small for the lower blocks. If we modify the model for a case where both sides of the coil are producing force we get the expected 2x increase:

Lorentz Force2.png
Calculation of force on wider coil

Now, to take those force numbers and get something useful out of them. The first thing we have to understand is the model. This is a planar model, which is basically a 2D case with some assumed thickness. The thickness doesn't affect any of the magnetic calculations, but it does affect the force calculation (the L term in the equation). KF made this model with 1mm thickness. So, we multiply the calculated force by the actual width of the magnets (20 mm), then use the average radius (90 mm) to calculate torque.

With those numbers I get:
0.0706N * 20 * 0.090m = 0.127 N-m torque for the 2nd case (both coils producing torque)
0.0326N * 20 * 0.090m = 0.059 N-m torque for the original case

If memory serves, the current design was 10 coils/phase, so total torque would be ~1.3 N-m. A little comparison shows that is nowhere near the 33.9 N-m design torque. Something seems amiss in either calculations or model...
Eric

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rhitee05
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Re: Doing the Math

rhitee05 wrote:There was only one thing wrong with your model - direction. You modeled the coil correctly, but on one side the number of turns needs to be negative to indicate the opposite current direction. <snip>

D'oh! Thank you for pointing this out

With the first plot, other than delete the materials for the rest of the model, what were the precise steps you took to model the coils?

Also thanks for spotting my units issue; my bad

rhitee05 wrote:<snip> If memory serves, the current design was 10 coils/phase, so total torque would be ~1.3 N-m. A little comparison shows that is nowhere near the 33.9 N-m design torque. Something seems amiss in either calculations or model...

I think I can help with that. LetÃ¢â‚¬â„¢s accept the reported value:
0.0706N * 20 * 0.090m = 0.127 N-m torque for the 2nd case (both coils producing torque)
And that there are 10 coils/phase; 10 * 0.127 N-m = 1.27 N-m / phase.

From Plan-D
32 Poles / 16:1, 30 Teeth / 10 teeth per phase
F = Ãâ€ž / r = 33.9 / 0.090 m = 377 N
Wheel spins at 7 rps.
@ 16:1 & 7 rps, the Frequency (f) = 7 * 16 = 112 Hz
The original problem has Current (I) = 44 A.
We calculated that the actual Current/Phase = I/Ã¢Ë†Å¡3 = 44 A / 1.732 = 25.4 A / single phase of a 3-phase circuit.
Therefore 25.4 A / 112 = 0.227 A / phase
However Ã¢â‚¬â€œ we changed the original radius from 0.3048 m to 0.09 m for Plan-D, a factor of 3.387. Therefore the new phase current is
0.227 * 3.387 = 0.769 Ã¢â€°Ë† 0.8 A
This is the actual value applied to the coils in the model; math is validated.

What happens if we reverse the math for Torque / phase?
Given 1.27 N-m / phase, @ 112 Hz * 3-phases =>
1.27 * 112 * Ã¢Ë†Å¡3 = 246 N-m applied Torque per second

Would this not imply that our magnetic utilization is very good?
What did we miss?
~KF
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Re: Doing the Math

Kingfish wrote:With the first plot, other than delete the materials for the rest of the model, what were the precise steps you took to model the coils?

Changing the magnets to non-magnetic material and correcting the coil so the lower block has -33 turns were the only changes. Run the analysis and the resulting field will be that solely due to the current.

Kingfish wrote:From Plan-D

32 Poles / 16:1, 30 Teeth / 10 teeth per phase
F = Ãâ€ž / r = 33.9 / 0.090 m = 377 N
Wheel spins at 7 rps.
@ 16:1 & 7 rps, the Frequency (f) = 7 * 16 = 112 Hz
The original problem has Current (I) = 44 A.
We calculated that the actual Current/Phase = I/Ã¢Ë†Å¡3 = 44 A / 1.732 = 25.4 A / single phase of a 3-phase circuit.
Therefore 25.4 A / 112 = 0.227 A / phase
However Ã¢â‚¬â€œ we changed the original radius from 0.3048 m to 0.09 m for Plan-D, a factor of 3.387. Therefore the new phase current is
0.227 * 3.387 = 0.769 Ã¢â€°Ë† 0.8 A

This is the actual value applied to the coils in the model; math is validated.

What happens if we reverse the math for Torque / phase?

Given 1.27 N-m / phase, @ 112 Hz * 3-phases =>
1.27 * 112 * Ã¢Ë†Å¡3 = 246 N-m applied Torque per second

Would this not imply that our magnetic utilization is very good?
What did we miss?

Ah, now I think I see where the issue lies. The torque production and the electrical frequency are unrelated! We can show this using dimensional analysis - frequency is not a unit-less quantity - Hz is 1/s, so if you take amps and divide by Hz, now you get A-s which is not what you want. Similarly, if you take N-m and multiply by Hz, you get N-m/s - I'm not even sure what that is, but it ain't torque! Sorry I didn't catch this sooner.

Let's back up to the current calculation. I agree with you to the point where we say 44 / sqrt(3) = 25.4 A per phase. All of these should be RMS values, so no further correction is necessary. If we take that current value and plug it into the FEMM model, and change the model depth to the proper 20 mm, we get a J x B force of 44.5 N, or 445N total over the 10 poles, or finally 770.8N for all 3 phases with the factor of sqrt(3). That works out to 69.4 N-m torque. That's significantly more than the initial design value (about 104% more), but this is a dual-stator design and the flux density is ~0.75T vs. the 0.5T value used in the prior calculations.

A much better result.
Eric

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Re: Doing the Math

Yea!

OK, I clearly see that I am still stuck in the sandbox with my maths - but at least there is a rainbow on the horizon

This sound to me that we are likely on par with 3 mm magnets, and much better with 4 mm then - and just about on target.

Let me get to fiddling about then and see if I can replicate your steps.

Mucho pleased & many thanks! KF
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Re: Doing the Math

Another very useful thing to do with modeling is to figure out the structural forces due to the magnetic fields. An axial-flux motor has a powerful tendency for the two rotors to crush together due to magnetic attraction. The structure of the motor has to withstand this, and in particular the rotors must have thrust bearings or other appropriate measures. We can use FEMM to estimate this force for the structural design.

First, some theoretical background. It's not a trivial task to calculate force due to two permanent magnets. There are formulas for simple cases, like two dipole magnets in free space. There are no formulas for something like a motor, but there is a general technique which can be used to find the force indirectly. We can use basic physics to find that F = - grad(Wm), where Wm is the stored energy in the magnetic field and grad() is the gradient (directional derivative).

I can illustrate this using a simple example. Lifting an object off the ground gives it some gravitational potential energy. We need to do some work in order to lift the object, and simple conservation of energy tells us that the increase in potential energy is equal to the work done, where work is E = F * d, force times distance. We can apply the above principle to find the force. We know that potential energy is U = m*g*h. To find the vertical force, we find dU/dh, which gives us F = - dU/dh = - m*g. This is a fairly obvious result - we already know the force on the ball is its weight (mass * gravity) in the downward direction. But this illustrates how our calculation will work.

In the motor, there is energy stored in the magnetic field (ignoring the windings for now, speaking only of the permanent magnets). Specifically, we can look at the energy stored in the fields contained within the motor airgap (note that in the coreless case this includes the entire volume between the rotors). If the magnets are pressed directly together, there is no airgap; this is the zero-potential case, just like an object sitting on the ground. As we pull the magnets apart, the potential increases, and by finding the derivative of the stored energy we can find the force. In some cases this can be done analytically, but in our case we will use FEMM and approximate the derivative using the limit definition: limit dWm/dx = (Wm(x+dx) - Wm(x)) / dx, as dx -> 0. That is, we can find the stored energy Wm for our model, Wm(x), then we will shift the element of interest (one rotor) very slightly and find it again, Wm(x+dx), then we can take the difference and divide by the shift dx. As dx approaches zero the result will approach the true value of the derivative. Since we are using a numerical model with limited resolution, we can't make dx too small or error will swamp the useful signal.

... and I'm going to leave it there for now. I don't believe the numbers I got from the first set of calculations, so I want to do a little verification before I post any numbers here. It might be a few days before I have time to get around to that. In the meantime, here's a Google Books link which explains the above:

Eric

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Re: Doing the Math

BeamBoy Studies - Part I

IÃ¢â‚¬â„¢ve been off-line on this thread for a while and actually working in the background with Eric on the FEMM calculations, and he is correct that we must account for the structural forces on our motor design.

After FEMM another tool that I find useful is BeamBoy: A Beam Analysis Tool. I have version 2.2 which is easily found through an Internet search. I donÃ¢â‚¬â„¢t have access to Mathematica and at \$10k a seat I am in no hurry to purchase a license Ã¢â‚¬â€œ however it is the software of choice for generating proper calculus. In the meantime we poor folk must eat with plasticware Ã¢â‚¬â€œ and BeamBoy - though lightweight, does a good job at forking over the data.

Breaking down model for study:
• Exterior Rotor faces (covers)
• Magnets
• Internal Rotor

Observations:
• The forces of attraction on the Internal Rotor with magnets nearly cancel out (though one may wish to study the shear forces internally). The calculations suggested deflections were less than 5% of 1 mm per side.
• The forces between the opposing Halbach magnets are weaker than the forces between the Internal Rotor and the External Rotor magnets. We will address this momentarily.
To study the model with our limited tools, we analyze the cross-section of the forces acting upon the External Rotor through one magnetic pole. It helps that we are a coreless design which adds economy to the calculations.

Presumptions:
• Generally, in the United States our supply of aluminum plate is cheaper if ordered in inches. Therefore I used two sizes of plate for the study: 0.125-in/3.17mm and 0.1875-in/4.76mm.
• In a previous correspondence between Eric and I the forces of attraction were suggested to be 78N.

Instructions for BeamBoy:
1. On Launch, select metric units and a Beam Length (Outside R) = 0.124 m
2. Add Simple Supports at 0.1165 m and at 0.024 m (theoretical points between the Bearing OD and the Fastener Bolt Circle)
3. Add Point Load at 0.090 m (Plan-D midpoint) of 78 N
1. Modulus of Elasticity = 71.7 GPa (Aluminum 7075-T6XX)
2. Select Calculate:
1. Rectangle
2. Solid
3. Width = 28 mm (2 magnets width for Halbach * 14 mm)
4. Height = 4.7625 mm = 0.1875 inch (typical)
5. Select Calculate, Ã¢â‚¬Å“Use these valuesÃ¢â‚¬Â, Continue
3. Calculate|Calculate Stress and Deflection (Ctrl-R)
5. Bottom Diagram: Deflection at the OD of the Hub is 0.178 mm.

I think we can live with 0.2 mm of deflection.

What happens if we twiddle with a couple of parameters though?

Hmmm, KF
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Re: Doing the Math

BeamBoy Studies - Part II

Curious about the potential forces for various magnet heights, I performed a two-track analysis across a spectrum of dimensions.

Borrowing from earlier Halbach calculations I developed a spreadsheet which provided the essential theoretical data. Then I developed slightly more elaborate FEMM studies to provide the measured data. Using this approach we get a sense of the high and low margins Ã¢â‚¬â€œ and the limits across the material heights, all which is useful for evaluating the structural behavior.

Figure 24 Explained:
The top-most formula is used to derive the theoretical peak flux density for a Halbach array. Much earlier in the thread I had considered using Â¼-inch N52 cube magnets and I simply grabbed a copy of those rough calcs. To the right of that list is the study continued with various magnet thicknesses from 3 to 12 mm. The two rows of conditional formatting below that are:
• Peak Flux Density in Tesla (T) from low to high.
• The difference between present and prior result.
It is easy to observe that the economic benefits are slowly reduced as height of the material increases.

The middle formula calculates the Force (F) of attraction between opposing magnetic faces. To simplify my Excel calculations I split the task into two parts before totaling. The Theoretical is given first: This is the Force of attraction between the entire distance (internal rotor excluded) between the two Halbach arrays.

From the adjusted FEMM models, I integrated along a line passing through the cross-section of the flux corner-to-corner between the External Magnet-Pole and the Internal Magnet-Pole, and this value is registered as Average Measured Flux Density. Recalculating using the FEMM data, we can see that the Forces are stronger between measured poles than between the Halbach arrays.

I noted a trend and went further to calculate the difference between the theoretical and the measured Flux Density and the values suggest a constant derivation or error; something worth noting.

Choosing the worst case:
The thing about engineering is this: Never assume your model is absolutely correct; you know... shit happens. I have been considering the 4 mm tall magnets as the best economic solution, although going to 5 mm has merit as well. If I won the Lotto Ã¢â‚¬â€œ sure 6 mm would even do, however 8 mm is out of reach. Now that I have agreed not to build an 8 mm, letÃ¢â‚¬â„¢s use it for the ultimate material stress.

Using the Instruction for BeamBoy in Part I with a 4.7625 mm thick beam and 138 N Force, the deflection from the center is 0.315 mm at the point of where the hub spokes attach. We can do a lot to reduce that value, namely by carving the External Rotor out from a thicker plate and employing ribbing, leaving the 4.7625 mm thick part at the magnet backing; anything to shorten that unreinforced thin section. It should be noted that 0.315 mm deflection is pretty dang good because the actual deflection at the magnet face where the coils are located is about 0.2 mm which is what we really care about.

Going one step farther Ã¢â‚¬â€œ just for peace of mind Ã¢â‚¬â€œ using the same model I changed the Force to 300 N: The deflection where the spokes mount is close to 0.7 mm, and at the magnet-coil surface approaches 0.5 mm or nearly 50% of our air gap. Without optimization, with the wildest magnetic forces applied to a motor that likely cannot be assembled, the structural integrity of the design should still allow the wheel to spin Ã¢â‚¬â€œ in theory.

Neat, huh? KF
Last edited by Kingfish on Fri Mar 04, 2011 3:17 pm, edited 1 time in total.
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Kingfish
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