lab electromagnet from scratch
 Rich Feldman
 Posts: 967
 Joined: Mon Dec 21, 2009 11:59 pm
 Real name: Rich Feldman
 Location: Santa Clara County, CA, USA
Re: lab electromagnet from scratch
In case the pictures are misleading: Yoke parts in BTI have areas that are 113% and 122% of the pole area, counting both legs. So yoke metal is always farther from saturation than pole metal. I think the corresponding area ratio in the Mullins magnet (MCM?) is 95% for both part types.
The exceptionally slender aspect ratio comes from 1) need to reach around two coils with >10K ampere turns each. That's the minimum to get 1 tesla in 1 inch of air, for any pole diameter. Plus roughly 6% to magnetize about 1 lineal meter of steel to similar B value, same as in MCM. And 2) keeping the pole area down to facilitate DIY steel fabrication and handling, since no minimum diameter was required. I still believe this would be hunky dory if not for the leakage flux challenge.
Thanks for all the hints. After some reading, thinking, and just a little computing, my grasp of leakage flux is much better than it was a few days ago. The Internet teaches that all electromagnets and transformers have leakage flux percentages. It helps to keep air gaps narrow (duh!). And if wide, to place the coils as close as possible to the gap.
Some simulations, and multipoint flux measurements (with various gap lengths), are in the planning stage.
I bet they will support the view that in this application, aspect ratios are what matter most. Permeability is second, and nonlinearity (saturation) is last.
Let's assume the core's top half is a mirror image of the bottom. Take the magnet's nominal size (pole diameter) as the unit of length. I claim that the dimensions which matter most are air gap length, pole length, and radial distance between pole and "side bar". The last two are driven by coil length and diameter. The numbers for MCM appear to be in the mainstream  about (0.24, 1, 1). For BTI they are (0.33, 2.33, 1.5).
For simplicity, keep B enough below saturation that the steel BH curve is still sort of linear. Then we can look at the B/2 field generated by bottom coil only. Later superimpose a mirror image to get the total field.
Consider the flux impelled upward by the lower coil, when it reaches the bottom pole surface. If there's no air gap, the least reluctant return path is to carry on into the upper pole piece, and come back around the yoke. Nothing but steel! Total reluctance is low, and so is the total MMF for a design amount of flux. The "magnetic potential" is distributed around the whole circuit, so a small fraction of the flux will take a short cut through air from the pole pieces to the yoke.
It doesn't take much of an air gap to greatly raise the magnetic potential difference between the two pole surfaces, and demand lots more ampere turns. Now as the flux emerges from lower pole, the upper pole surface isn't so inviting. The yoke bars are still on the far side of some air, but their broadside view presents plenty of area.
I think simulations (and measurements) will show that when the air gap reaches 1 inch, between 3 inch diameter poles, the total reluctance of sideways leakage paths is less than that of the air gap itself. Both are much larger than the reluctance of the intended flux path through steel. Yoke bars could all be twice as thick, or twice as permeable, without greatly reducing the leak percentage. If 3D simulation were available, we might see if it helps to place my side plates with narrow edges instead of broad faces oriented toward the pole axis.
And that's one man's novice opinion.
The exceptionally slender aspect ratio comes from 1) need to reach around two coils with >10K ampere turns each. That's the minimum to get 1 tesla in 1 inch of air, for any pole diameter. Plus roughly 6% to magnetize about 1 lineal meter of steel to similar B value, same as in MCM. And 2) keeping the pole area down to facilitate DIY steel fabrication and handling, since no minimum diameter was required. I still believe this would be hunky dory if not for the leakage flux challenge.
Thanks for all the hints. After some reading, thinking, and just a little computing, my grasp of leakage flux is much better than it was a few days ago. The Internet teaches that all electromagnets and transformers have leakage flux percentages. It helps to keep air gaps narrow (duh!). And if wide, to place the coils as close as possible to the gap.
Some simulations, and multipoint flux measurements (with various gap lengths), are in the planning stage.
I bet they will support the view that in this application, aspect ratios are what matter most. Permeability is second, and nonlinearity (saturation) is last.
Let's assume the core's top half is a mirror image of the bottom. Take the magnet's nominal size (pole diameter) as the unit of length. I claim that the dimensions which matter most are air gap length, pole length, and radial distance between pole and "side bar". The last two are driven by coil length and diameter. The numbers for MCM appear to be in the mainstream  about (0.24, 1, 1). For BTI they are (0.33, 2.33, 1.5).
For simplicity, keep B enough below saturation that the steel BH curve is still sort of linear. Then we can look at the B/2 field generated by bottom coil only. Later superimpose a mirror image to get the total field.
Consider the flux impelled upward by the lower coil, when it reaches the bottom pole surface. If there's no air gap, the least reluctant return path is to carry on into the upper pole piece, and come back around the yoke. Nothing but steel! Total reluctance is low, and so is the total MMF for a design amount of flux. The "magnetic potential" is distributed around the whole circuit, so a small fraction of the flux will take a short cut through air from the pole pieces to the yoke.
It doesn't take much of an air gap to greatly raise the magnetic potential difference between the two pole surfaces, and demand lots more ampere turns. Now as the flux emerges from lower pole, the upper pole surface isn't so inviting. The yoke bars are still on the far side of some air, but their broadside view presents plenty of area.
I think simulations (and measurements) will show that when the air gap reaches 1 inch, between 3 inch diameter poles, the total reluctance of sideways leakage paths is less than that of the air gap itself. Both are much larger than the reluctance of the intended flux path through steel. Yoke bars could all be twice as thick, or twice as permeable, without greatly reducing the leak percentage. If 3D simulation were available, we might see if it helps to place my side plates with narrow edges instead of broad faces oriented toward the pole axis.
And that's one man's novice opinion.
Richard Feldman
 Rich Feldman
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 Joined: Mon Dec 21, 2009 11:59 pm
 Real name: Rich Feldman
 Location: Santa Clara County, CA, USA
Re: lab electromagnet from scratch
Been talking with Chris, but haven't yet updated the MCM coil dimensions from guesses in this picture.
Discovered a cool way to name the nodes in onedimensional magnetic path model. In the round parts, blue lines are drawn where they bisect the semicircle area. Chris, should the NS order in MCM figure be flipped?
Sketch at the top left shows axisymmetric model of BTI for the first FEMM simulations. Left edge is the axis of rotation. Yoke horizontal and vertical bars are represented by disks and a thick tube with the same crosssectional areas (still 113% and 122% of pole area). So the model is configured like a gapped ferrite pot core. I bet it will overestimate leakage flux.
Between the poles, FEMM model has a few very short cylinders that can be steel or air. Here there's a 0.1 inch air gap, which gives a nice qualitative view of leakage flux. Only the bottom coil (with practice coil dimensions and turns count) is energized. That makes it easier to understand the leakage flux behavior. If we were to add an identical top coil, it would superimpose a mirrorimage field distribution, restoring top/bottom symmetry at the gap. Since BTI's design target doesn't specify the strong field diameter, I'm thinking about tapered pole ends. Time to stop pinching pennies so hard, and shop out some of the fabrication work. Cost to date is under $100 for all parts in the picture, including extension cord and four casters. I've probably spent more time talking about it than working on it.
Discovered a cool way to name the nodes in onedimensional magnetic path model. In the round parts, blue lines are drawn where they bisect the semicircle area. Chris, should the NS order in MCM figure be flipped?
Sketch at the top left shows axisymmetric model of BTI for the first FEMM simulations. Left edge is the axis of rotation. Yoke horizontal and vertical bars are represented by disks and a thick tube with the same crosssectional areas (still 113% and 122% of pole area). So the model is configured like a gapped ferrite pot core. I bet it will overestimate leakage flux.
Between the poles, FEMM model has a few very short cylinders that can be steel or air. Here there's a 0.1 inch air gap, which gives a nice qualitative view of leakage flux. Only the bottom coil (with practice coil dimensions and turns count) is energized. That makes it easier to understand the leakage flux behavior. If we were to add an identical top coil, it would superimpose a mirrorimage field distribution, restoring top/bottom symmetry at the gap. Since BTI's design target doesn't specify the strong field diameter, I'm thinking about tapered pole ends. Time to stop pinching pennies so hard, and shop out some of the fabrication work. Cost to date is under $100 for all parts in the picture, including extension cord and four casters. I've probably spent more time talking about it than working on it.
Richard Feldman
 Rich Feldman
 Posts: 967
 Joined: Mon Dec 21, 2009 11:59 pm
 Real name: Rich Feldman
 Location: Santa Clara County, CA, USA
Re: lab electromagnet from scratch
Couldn't resist posting more axisymmetric simulation results, before trying a planar model of the same real object.
When there's no air gap, there's practically no leakage flux. 1D model would be quite accurate. The new chart is B along a contour from bottom to top, at half of the pole radius. Current is 1 ampere in the extension cord coil. With a halfinch air gap, the peak B is vastly smaller (as predicted by 1D model), and gap B is even smaller by another factor of three. Let's see what happens when we increase the vertical yoke bar area from 1.22 to 2.35 times the pole area. Not much, eh? Tends to confirm that this is a pole aspect ratio thing. Of course we need to repeat the experiment on a traditional, stubby geometry like MCM. Maybe Chris will beat me to it.
When there's no air gap, there's practically no leakage flux. 1D model would be quite accurate. The new chart is B along a contour from bottom to top, at half of the pole radius. Current is 1 ampere in the extension cord coil. With a halfinch air gap, the peak B is vastly smaller (as predicted by 1D model), and gap B is even smaller by another factor of three. Let's see what happens when we increase the vertical yoke bar area from 1.22 to 2.35 times the pole area. Not much, eh? Tends to confirm that this is a pole aspect ratio thing. Of course we need to repeat the experiment on a traditional, stubby geometry like MCM. Maybe Chris will beat me to it.
Richard Feldman
 Rich Feldman
 Posts: 967
 Joined: Mon Dec 21, 2009 11:59 pm
 Real name: Rich Feldman
 Location: Santa Clara County, CA, USA
Re: lab electromagnet from scratch
Tried insulating the sideways leakage path with a sheet of superconductor. Set u = 0.001; maybe zero would work. Pattern changes, but the flux still finds ways to leak around the desired gap.turn a few pounds of steel into swarf.
Then came a planargeometry model of BTI. Yoke part widths are fudged differently, to keep the crosssectional area ratios right. All flux is parallel to the plane of the paper. (As in the axisymmetric model, where the plane is any that includes the axis.) I think the planar model underestimates leakage and fringing flux; the axi. model would be good about fringing but very pessimistic about sideways leakage. The planar problem size could be halved, with the right boundary condition applied at the line (plane) of bilateral symmetry. The round shells were set up by a FEMM wizard, as a boundary condition to emulate unbounded space.
This calls for lab measurements. I've wound a round fluxmeter sense coil to fit around pole pieces. Got the bobbin made for a rectangular one, to fit yoke side bars. Sensitivity calibration is easy & very accurate. Rectangular coil will have 10.00 +/ 0.01 turns. Round coil also has 10 turns, with a tap at 5 turns.
The tap will allow direct comparison of flux in one sidebar with half the flux in a pole piece, without changing the instrument range or doing arithmetic. Come to think of it, the sense coils could be connected in series to read the difference between pole and sidebar fluxes. Then slid vertically to find null places (height pairs where pole and side fluxes are equal). No fancy voltage integrator needed for that!
A simulation with tapered pole tips also revealed little improvement, a finding that initially came as a surprise. Needs more review & thought, but much easier than having to Then came a planargeometry model of BTI. Yoke part widths are fudged differently, to keep the crosssectional area ratios right. All flux is parallel to the plane of the paper. (As in the axisymmetric model, where the plane is any that includes the axis.) I think the planar model underestimates leakage and fringing flux; the axi. model would be good about fringing but very pessimistic about sideways leakage. The planar problem size could be halved, with the right boundary condition applied at the line (plane) of bilateral symmetry. The round shells were set up by a FEMM wizard, as a boundary condition to emulate unbounded space.
This calls for lab measurements. I've wound a round fluxmeter sense coil to fit around pole pieces. Got the bobbin made for a rectangular one, to fit yoke side bars. Sensitivity calibration is easy & very accurate. Rectangular coil will have 10.00 +/ 0.01 turns. Round coil also has 10 turns, with a tap at 5 turns.
The tap will allow direct comparison of flux in one sidebar with half the flux in a pole piece, without changing the instrument range or doing arithmetic. Come to think of it, the sense coils could be connected in series to read the difference between pole and sidebar fluxes. Then slid vertically to find null places (height pairs where pole and side fluxes are equal). No fancy voltage integrator needed for that!
Richard Feldman

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 Joined: Sat Jun 03, 2017 4:26 am
 Real name: Mark Kimball
Re: lab electromagnet from scratch
If you are not already using it, I have found FEMM's scripting capability (via LUA) to be useful for playing with different magnet configurations. It can speed things up quite a bit compared to manually setting dimensions of your parts (or coil currents etc.). It can take a little time to write functions to create components with rectangular and cylindrical profiles, but after that it is much less painful to experiment.
Mark
Mark

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 Joined: Sun Feb 19, 2017 4:32 pm
 Real name: Chris Mullins
 Location: Shenandoah Valley, VA
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Re: lab electromagnet from scratch
Mark, I can second the recommendation for FEMM's Lua scripting  it was very easy to that working.
I simulated the "MCM", as Rich has named it, this weekend. Two immediate observations: 1) FEMM works pretty well, and 2) I should have simulated the MCM before actually building it.
I drew a 2D slice of the MCM, not accounting for the roundness of the pole piece. My first question was how the field in the gap varies with current, especially considering my plans to upgrade from the 3kW (55A) I have now to 5kW and even 10kW (75 or 100A of coil current). My original design calculations used the simple formula of B = 4*pi*N*I/G, where B is field strength (Teslas), N is number of coil turns, I is current, and G is gap width. Clearly that will overestimate the field when the iron starts saturating, but I didn't simulate that. At my full current of 55A, I get this from FEMM: which shows about 860 mT in the gap, compared to 976 mT predicted by the simple formula.
Using Lua, I scripted the coil current ramping from 5A to 100A in 5A increments, and extracted the field at the center of the pole gap. The script is pretty simple:
Extracting those results from the console and plotting them along with the simple formula shows that even at 55A, the efficiency is only 85%, which is close to what I actually measured, and at 100A, it drops to 66%. My son and I set up our gauss meter as close to we could in the center of the gap, and stepped the current from 5A to 55A (as high as we can go), and plotted those on the same graph. The results match FEMM very closely:
The red line is the simple formula value, the blue is the FEMM output, and the green diamonds are our actual measurements. At 100A (10kW) I'd only get 1.16 Tesla, which isn't worth the effort (would take a single phase to 3 phase converter, a much larger power supply, and much better cooling). Even 5kW doesn't get very far.
The FEMM vs. measured efficiencies (ratio of actual to predicted field strength) are pretty close: I would expect the physical magnet to be worse than the simulation, given nonideal frame flatness, small gaps, etc., and generally it was a little lower (12%). Some of the difference is likely measurement error too. We started by removing any remanence from the frame (typically about 15 mT), then stepping from 0A up to 55A, but our current measurement resolution was only 0.1A.
Switching the frame from 1018 steel (what I actually used) to 1006 in FEMM improves things a bit  FEMM gives me 920 mT instead of 860 mT (compared to 976 in the formula). I couldn't readily find a low cost source for any lower carbon steel than 1018 though.
Although I have both coils installed now, for a while I was running with just the bottom coil, and had noticed a difference in field strength from the face of the bottom pole to the top pole. This difference varied with position (almost none at the pole center, getting worse towards the pole edge). I simulated this with FEMM, and that also agreed with what I had seen. I used a Lua script to measure the field from pole center out to the edge, in 0.1" increments, at both the top and bottom of the gap: When I had noticed this effect earlier I noted a 25% difference, as measured roughly an inch or so in from the pole I (that wasn't very controlled; I was pushing my field sensor as far as it could reach in the gap between my chamber and the pole). That roughly agrees with the FEMM result. As Richard pointed out, a 72 Gauss difference out of 1449 isn't much, and could be from the asymmetry of just one coil (longer flux path), or tiny imperfections in the frame, etc. I'm sure both effects are there to an extent, but the FEMM result suggests the former is dominant here, within my measurement error. I considered disconnecting the upper coil and taking more careful measurements, but didn't get that far.
Next, I looked at the field (with both coils) at the center of the gap (vertically), as it varied with distance from the pole center to the edge. This is where any pole shape optimization would be needed to improve the cyclotron. I didn't have a quick method to vary the field probe radially from the center in a controlled, measured manner, so I don't have physical measurements to compare with. Here's the FEMM output though, from another Lua script that steps through the field measurements: Finally, in another discussion I mentioned attempting to measure the "DC" inductance of the magnet:
Rich, thanks for the suggestion on FEMM  it's definitely a powerful tool, and not difficult to use once you know what to do. In case it's helpful for anyone else getting started, here's a link to the MCM FEMM model: https://mullinscyclotron.weebly.com/upl ... _2coil.fem
Maybe a brief overview of FEMM (including scripting) for electromagnet modelling would be a good post for the new magnetics FAQ?
I simulated the "MCM", as Rich has named it, this weekend. Two immediate observations: 1) FEMM works pretty well, and 2) I should have simulated the MCM before actually building it.
I drew a 2D slice of the MCM, not accounting for the roundness of the pole piece. My first question was how the field in the gap varies with current, especially considering my plans to upgrade from the 3kW (55A) I have now to 5kW and even 10kW (75 or 100A of coil current). My original design calculations used the simple formula of B = 4*pi*N*I/G, where B is field strength (Teslas), N is number of coil turns, I is current, and G is gap width. Clearly that will overestimate the field when the iron starts saturating, but I didn't simulate that. At my full current of 55A, I get this from FEMM: which shows about 860 mT in the gap, compared to 976 mT predicted by the simple formula.
Using Lua, I scripted the coil current ramping from 5A to 100A in 5A increments, and extracted the field at the center of the pole gap. The script is pretty simple:
Code: Select all
showconsole()
mydir="./"
open(mydir .. "mcm_2coil.fem")
mi_saveas(mydir .. "temp.fem")
clearconsole()
for n=0,100,5 do
mi_modifycircprop("bottom coil",1,n)
mi_modifycircprop("top coil",1,n)
mi_analyze()
mi_loadsolution()
A, B1, B2 =mo_getpointvalues(0,10.75)
bmag=sqrt(B1*B1 + B2*B2)
print(n,bmag)
end
mo_close()
mi_close()
The FEMM vs. measured efficiencies (ratio of actual to predicted field strength) are pretty close: I would expect the physical magnet to be worse than the simulation, given nonideal frame flatness, small gaps, etc., and generally it was a little lower (12%). Some of the difference is likely measurement error too. We started by removing any remanence from the frame (typically about 15 mT), then stepping from 0A up to 55A, but our current measurement resolution was only 0.1A.
Switching the frame from 1018 steel (what I actually used) to 1006 in FEMM improves things a bit  FEMM gives me 920 mT instead of 860 mT (compared to 976 in the formula). I couldn't readily find a low cost source for any lower carbon steel than 1018 though.
Although I have both coils installed now, for a while I was running with just the bottom coil, and had noticed a difference in field strength from the face of the bottom pole to the top pole. This difference varied with position (almost none at the pole center, getting worse towards the pole edge). I simulated this with FEMM, and that also agreed with what I had seen. I used a Lua script to measure the field from pole center out to the edge, in 0.1" increments, at both the top and bottom of the gap: When I had noticed this effect earlier I noted a 25% difference, as measured roughly an inch or so in from the pole I (that wasn't very controlled; I was pushing my field sensor as far as it could reach in the gap between my chamber and the pole). That roughly agrees with the FEMM result. As Richard pointed out, a 72 Gauss difference out of 1449 isn't much, and could be from the asymmetry of just one coil (longer flux path), or tiny imperfections in the frame, etc. I'm sure both effects are there to an extent, but the FEMM result suggests the former is dominant here, within my measurement error. I considered disconnecting the upper coil and taking more careful measurements, but didn't get that far.
Next, I looked at the field (with both coils) at the center of the gap (vertically), as it varied with distance from the pole center to the edge. This is where any pole shape optimization would be needed to improve the cyclotron. I didn't have a quick method to vary the field probe radially from the center in a controlled, measured manner, so I don't have physical measurements to compare with. Here's the FEMM output though, from another Lua script that steps through the field measurements: Finally, in another discussion I mentioned attempting to measure the "DC" inductance of the magnet:
FEMM can give the total stored energy in a magnetic field, and running that for the MCM at 55A gives 81 Joules. Working from E=0.5*L*I^2, the inductance is around 54 mH, somewhere between the 100Hz meter value of 34 mH and my (very) rough 80 mH. I suspect FEMM is closer than my estimate ....Total coil inductance is difficult to measure directly. My LCR meter only goes to 100 Hz, not low enough to get the "DC" inductance. Readings at 100, 120, and 1000 Hz are 34.3, 32.9, and 14.6 mH, respectively. A (very) rough L/R time constant measurement going from zero to 12 amps gives around 80 mH
Rich, thanks for the suggestion on FEMM  it's definitely a powerful tool, and not difficult to use once you know what to do. In case it's helpful for anyone else getting started, here's a link to the MCM FEMM model: https://mullinscyclotron.weebly.com/upl ... _2coil.fem
Maybe a brief overview of FEMM (including scripting) for electromagnet modelling would be a good post for the new magnetics FAQ?

 Posts: 20
 Joined: Sat Jun 03, 2017 4:26 am
 Real name: Mark Kimball
Re: lab electromagnet from scratch
Chris,
Wow, you really got all over that one! It is good to know that FEMM does a pretty good job of simulating electromagnets. Hopefully it's as good with NdFeB magnets because that's what I'm interested in.
Mark
Wow, you really got all over that one! It is good to know that FEMM does a pretty good job of simulating electromagnets. Hopefully it's as good with NdFeB magnets because that's what I'm interested in.
Mark

 Posts: 31
 Joined: Sun Feb 19, 2017 4:32 pm
 Real name: Chris Mullins
 Location: Shenandoah Valley, VA
 Contact:
Re: lab electromagnet from scratch
Yeah, well I did have a long weekend to work on it
Rich pointed out offline that I may have two flaws in my FEMM model that could be roughly canceling each other out, so that the results happen to be close to correct on some of those results. I'll try the axisymmetric model this weekend, and varying my planar model to explore that more closely.
All goes to show it's important to have a good understanding of the underlying principles, how the models work, and the expected results when using simulations. Otherwise it may be "garbage in, garbage out."
Rich pointed out offline that I may have two flaws in my FEMM model that could be roughly canceling each other out, so that the results happen to be close to correct on some of those results. I'll try the axisymmetric model this weekend, and varying my planar model to explore that more closely.
All goes to show it's important to have a good understanding of the underlying principles, how the models work, and the expected results when using simulations. Otherwise it may be "garbage in, garbage out."
 Rich Feldman
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 Joined: Mon Dec 21, 2009 11:59 pm
 Real name: Rich Feldman
 Location: Santa Clara County, CA, USA
Re: lab electromagnet from scratch
Here's a drawing to illustrate the onedimensional "magnetic reluctance" model which Richard Hull mentioned here. Hats off to Oliver Heaviside.
I think it can explain most of the "inefficiency" when Chris's measurements & simulations fall short of the simple B = u0*N*I/G. That formula is based on reluctance of the air gap, without consideration of that in the steel path. The formula predicts 20212 ampereturns per tesla for a 1" air gap, but we need another 2248 ampereturns per tesla to magnetize the steel. The MCM and BTI electromagnet steel parts and 1" air gaps are rendered as rectangles. Height is proportional to physical length in the magnetic flux direction; width is proportional to the cross sectional area. (So the shaded area ratio matches the mass ratio, except for square corner details.)
By analogy with electrical resistance, we can figure the reluctance of each section in ampereturns per weber (like volts per amp).
It's equal to length, divided by cross sectional area, divided by the magnetic permeability. For air that's u0 = 4 pi / 10^7. In MS Excel, 4e7*pi(). For steel I used a value 500 times greater (more permeable). Ampereturns per weber is dimensionally the same as inverse henries. For wound cores like we're talking about here, I think the inductance constant L/N^2 is the inverse of the total reluctance  if all the flux linked all the turns.
The 8" magnet has reluctance values roughly 7 times smaller than the 3" magnet, because that's the pole area ratio.
But for a given B value, the 8" magnet needs 7 times more flux (in webers). So the associated MMF (magnetomotive force) values, in ampereturns, are roughly the same. They're identical for the 1" long air gaps and 7" long round pole pieces.
This model is accurate for very small air gaps. But it doesn't account for 2D and 3D effects: fringing flux near the gap and leakage flux between pole pieces and the yoke. I've learned that those are hugely significant for the slender magnet on the right side of picture.
I think it can explain most of the "inefficiency" when Chris's measurements & simulations fall short of the simple B = u0*N*I/G. That formula is based on reluctance of the air gap, without consideration of that in the steel path. The formula predicts 20212 ampereturns per tesla for a 1" air gap, but we need another 2248 ampereturns per tesla to magnetize the steel. The MCM and BTI electromagnet steel parts and 1" air gaps are rendered as rectangles. Height is proportional to physical length in the magnetic flux direction; width is proportional to the cross sectional area. (So the shaded area ratio matches the mass ratio, except for square corner details.)
By analogy with electrical resistance, we can figure the reluctance of each section in ampereturns per weber (like volts per amp).
It's equal to length, divided by cross sectional area, divided by the magnetic permeability. For air that's u0 = 4 pi / 10^7. In MS Excel, 4e7*pi(). For steel I used a value 500 times greater (more permeable). Ampereturns per weber is dimensionally the same as inverse henries. For wound cores like we're talking about here, I think the inductance constant L/N^2 is the inverse of the total reluctance  if all the flux linked all the turns.
The 8" magnet has reluctance values roughly 7 times smaller than the 3" magnet, because that's the pole area ratio.
But for a given B value, the 8" magnet needs 7 times more flux (in webers). So the associated MMF (magnetomotive force) values, in ampereturns, are roughly the same. They're identical for the 1" long air gaps and 7" long round pole pieces.
This model is accurate for very small air gaps. But it doesn't account for 2D and 3D effects: fringing flux near the gap and leakage flux between pole pieces and the yoke. I've learned that those are hugely significant for the slender magnet on the right side of picture.
Richard Feldman