Magnetic Circuit Design:

recoil operation.

by Dr. Peter Campbell

In the previous Section, we described the simple situation of a permanent magnet providing flux to a single air gap via two steel pole pieces, and showed that the specific dimensions of this magnet and this air gap would yield a unique load line. The permanent magnet material itself is characterized by its demagnetization curve, the intercept between the two giving an operating point with a unique combination of B_m and H_m for the magnet in this condition. Should the air gap length l_g increase, then the slope of the load line will fall and the magnet's operating point will change.

Now, if the air gap increases incrementally and then decreases again to its original value, even cycling through these values, the operating point will run down and up the demagnetization curve in the manner shown in the B vs. H diagram above. Now let's examine this behavior in a little more detail.

The demagnetization curve is a relationship between B_m and H_m including the magnetization M. Differentiating with respect to H_m shows that, if M is constant, then the slope of the demagnetization curve is µ_o, the permeability of free space. Actually, the slope being µ_o is synonymous with the magnet's M being constant, because any decline in M tends to accelerate with growing demagnetization. Just remember that a decline in M is due to a partial reversal of this magnetization, from which a magnet does not recover without remagnetization. Consequently, as long as the demagnetization curve is linear, this means that there are no irreversible losses and the operating point will run up and down the same characteristic reversibly as shown above.

Furthermore, in Section 1C we defined the remanence (when H_m=0) in an ideal magnet as B_r=µ_oM_sat. Actually, if a partial reversal of M occurs as a real magnet operates throughout the second quadrant of the B vs. H diagram, then there is really no reason to suppose that this did not also occur to some lesser degree in the first quadrant; i.e. after saturation, the magnet settles to a remanence at B_r=µ_oM_r, where M_r<M_sat.

In several earlier Sections, we have also shown that the most serious decline in M occurs as the operating point enters the "knee" of the demagnetization curve, because at this (or a greater) level of demagnetization the magnet suffers an irreversible loss of its magnetization due to more significant partial reversals. If the demagnetization exceeds the intrinsic coercivity -H_ci, then the reversal of M is complete; more typically though, a magnet may enter the region of the "knee" and suffer a less critical reduction in M, but well below its M_sat level. As we have seen, H_ci is dependent upon the temperature, and so is the position of the "knee". For the anisotropic neodymium-iron-boron characteristics used in the example above, the "knee" encroaches the second quadrant as the magnet heats up above room temperature.

Now consider that a changing air gap cycles the load line through the same range as before, but with the magnet now operating at +60^oC rather than +20^oC. The diagram below shows that, at the largest l_g (lowest slope), the operating point passes the "knee" and the magnet suffers an irreversible loss of its magnetization. The operating point cannot return up the major demagnetization curve, but will follow a path within this characteristic. Since an irreversible loss has occurred, the magnet can only be returned to its original condition if it is fully remagnetized.

Recoil Operation

Remagnetization is generally not practical for a magnet that has been installed in a magnetic circuit, but if the loss of magnetization is not too great, a device may be designed to allow the magnet to operate within its major demagnetization curve at acceptably lower flux levels. The (black) line in the diagram above that the operating point follows to its original load line is called a recoil line, which is actually part of a minor magnetization curve. Recoil lines originating from various locations on a demagnetization curve are plotted on the diagram to the right. Notice that the excursion of the operating point in recoil does not track exactly up and back along a unique line, but the very narrow loops that are plotted are always assumed to be lines for calculation purposes. Also notice that, no matter where they originate, all recoil lines have approximately the same slope, and this is a characteristic of a permanent magnet material known as its recoil permeability, µ_rec. If a recoil line is projected to intercept the B_m axis at a reduced "remanence" of B_r=µ_oM_r, then the equation (similar to that for the demagnetization curve) describing it will be:

For operation in recoil, it is the recoil line (rather than the demagnetization curve) and the load line which are the two characteristic equations describing a particular magnet in a particular magnetic circuit, the intercept between which gives a unique magnet operating point with a specific B_m and H_m for the magnet. In terms of the B_m vs. H_m diagram from Section 4A (click the button at the top of this page to see this), a recoil line is substituted for the demagnetization curve.