This PerMag example illustrates the idea of a magnetic circuit and demonstrates the reasons for using iron in magnets. If you want to review the basic physics, see Sect. 5.7 in my book Principles of Charged Particle Acceleration available for free download at:
http://www.fieldp.com/cpa.html
Use these links to download the example input files:
http://www.fieldp.com/myblog/examples/magnetcircuit.min
http://www.fieldp.com/myblog/examples/magnetcircuit.pin
The first figure below shows a z-r plot of the geometry of a laboratory magnet. (Remember, in such a plot the parts are figures of revolution about the bottom axis.) The function of the device is to produce a fairly uniform field Bg = 0.5 tesla in the gap. We can estimate the required current NI amp-turns in each coil from Ampere’s law:
? (B/?r)*dl = 2*?0*NI, (1)
where the path of the circuit integral encloses the two coils. In the air gap, ?r = 1.0. In the iron, we assume that ?r = 500.0. If D is the width of the gap, we can rewrite Eq. 1 as
Bg*D + (1/500) ? B*dl = 2*?0*NI, (2)
where the integral in the second term part on the left-hand side is taken through the iron. Even though this path is longer, it makes only a small contribution because of the high value of ?r. The required current per coil is therefore about
NI ? Bg*D/(2*?0). (3)
For Bg = 0.5 tesla and D = 0.02540 m, we find that NI = 5040.0 A-turns.
Lines of magnetic flux density for the solution are shown in the top part of the first figure. The gap field near the axis is Bz = 0.4865 tesla, close to our estimate.
To understand the contribution of the iron, we can remove it and do a second solution. A quick way is to edit magnetcircuit.pin, commenting out the high value of ?r for the iron and substituting ? = 1.0:
* Region 2: IRON * Mu(2) = 5.0000E+02 Mu(2) = 1.0000E+00
The bottom of the first figure shows the resulting field lines. The second figure shows a plot of Bz(0,r) on the midplane of the gap from r = 0.0″ to r = 2.0″. Without iron, the average gap field drops to only 0.0701 tesla.
From an inspection of the figures, we can see the benefits of iron in the magnetic circuit:
- For a given coil current, we can achieve a much higher gap flux density. With iron, the material currents carry the magnetic flux over most of its transit while the coil current need support the flux only across the gap. For room temperature coils, the power dissipation to achieve a field of 0.5 tesla in the gap without iron would be about 50 times higher!
- Iron has a shielding effect so that flux density outside the magnet is very low.
- A high degree of field uniformity in the gap can be achieved by shaping the iron pole faces.
As an exercise, take line integrals in PerMag to get a numerically-exact value for the second term on the right-hand side of Eq. 2. Does the contribution account for the field difference: 0.4865 tesla versus 0.5000 tesla?
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