Counter-intuition in permanent magnet circuits

On average, physical intuition is useful half the time and misleading the other half. As a case in point, a trial user set up what he felt was the most basic magnet problem to test Magnum. The program returned results that differed from his expectation by about a factor of five. He contacted me to find what was wrong with his calculation. As it turned out, his setup was perfectly correct and the program was returning the right answer. The expectation was wrong.

The concept of a magnetic circuit is often useful to estimate achievable magnetic flux density in a laboratory magnet (i.e., a system with an iron flux conductor, a small air gap and a drive coil). The theory is described in Sect. 5.7 of my book Principles of Charged Particle Acceleration (available for free download at http://www.fieldp.com/cpa.html. The relationships follow from Ampere’s law and the conservation of magnetic flux. The issue is clouded for circuits driven by permanent magnets. In fact, I have to admit that Sect. 5.8 in my book is misleading. In reality, the gap field in a permanent magnet circuit depends on how the assembly is constructed.

Figure 1. Permag calculations of permanent magnetic circuits.

Figure 1. Permag calculations of permanent magnetic circuits.

The issue can best be understood with an example. Figure 1 shows a cylindrical magnetic circuit for a PerMag calculation. The air gap and permanent magnet are cylindrical with 2.0 cm radius. The gap length is Lg = 1.0 cm and the magnetic length is Lm = 2.0 cm. The ideal permanent magnet has Br = 1.0 tesla and the iron has μr = 500. The intuitive view of a magnetic circuit is that the lines of B are forced to flow in the high-reluctance iron (i.e., the iron is a magnetic conductor). In this case, the behavior of the circuit should not depend on the relative position of the permanent magnet and we expect the same gap field for the two geometries illustrated in Fig. 1. Under the assumption of contained flux, the theory in Sect. 5.8 predicts an average gap field:

Bg = Br/(1 + Lg/Lm)

For parameters of calculation, formula implies Bg = 0.67 tesla.

The PerMag calculation gives a gap field of about Bg = 0.57 tesla for case a. The result is consistent with theory if we consider that there are flux losses from leakage and fringing flux. On the other hand, the gap field for the bottom configuration is only 0.15 tesla, a significant difference. Clearly, the idea that B lines are “contained” by the iron is wrong. We need a better understanding of the psychology of flux lines.

A line of magnetic flux density emerging from the left-hand side of the permanent magnet wants to find its way to the right hand side the easiest way possible (i.e., the path of least reluctance). For case a, the B lines emerge directly into the air gap. Most of them follow the easiest path — cross the short gap and return through the iron. A fraction of the lines take a shortcut to the outer iron piece or return directly through air to the upstream side of the magnet.

In contrast, the B lines in case b must travel a relatively long distance through the iron stalk before reaching the air gap. The reluctance of the radial gap between the inner and outer iron pieces is less than that of the working gap. The result is that B lines cut out early and most of the flux never reaches the working gap. The example has an important practical implication. If the goal is to produce the maximum flux in an air gap of a permanent magnet circuit, the magnets should be adjacent to the gap.

For more information on Magnum and PerMag, please see http://www.fieldp.com/magnum.html and http://www.fieldp.com/permag.html.

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