(B/H) OF ROUND ROD MAGNETS IN AIR
George P. Gogue
Joseph J. Stupak, Jr.
Under some conditions, a magnet which has been magnetized in a fixture or coil, which is then placed in a magnetic circuit, is found to produce a weaker field than was calculated, based on its fully magnetized B/H curve. Another form of the same problem occurs when a magnet which has been magnetized in place, in a magnetic circuit, is found to produce its intended field initially but would become weaker if removed from the circuit and then returned to it. Magnets under these conditions are said to have undergone "shape demagnetization". The problem is much more severe with older Alnico materials which have a high flux-density B but relatively little coercivity H, and also a B/H curve which is more of a continuous arc than a straight line with a fairly sharp knee. Designers who have only worked with the newer magnet materials may not be aware that the danger of partial demagnetization by shape exists in these materials too.
II DISCUSSION OF SOLUTION:
It is very common in magnetic design to regard the magnet as being in one average single state of magnetization throughout. In fact both the flux-density B and the coercivity H (and hence, the ratio of B/H) may vary considerably from point to point within the magnet. This is especially true if the magnet is magnetized separately from the magnetic circuit and then is moved to the intended location, being temporarily exposed to open circuit conditions in air in between.
Figure 1: Flux around pole magnet
The nature of the problem can be understood intuitively by considering the flux paths of a plate magnet, as shown in Figure 1. Flux passes in a nearly straight line through the magnet and then through air, in an arc, to the other side. It is apparent that lines near the edges of the magnet have a shorter distance to travel than those further away. One would expect the longer path in air to have more external reluctance. If the magneto motive force (mmf) through the magnet were the same for both paths, then the flux-density should be less for the inner path than for the outer one. Therefore, with a material of B/H curve as shown in Figure 2, at the inner path, B would decrease, H would increase, and the B/H ratio would decrease at a faster rate than the decrease of B.
Figure 2: Ferrite (M8B) B/H curve at 20°C
The curve of Figure 2 has a "knee" and it drops off very quickly to the left of this region, with negatively increasing H. When in use, the B/H state of the magnet may vary due to magneto motive force contributions from nearby windings with electric current, or by changes of gap, temperature effects, etc. As long as the excursions of the operating point (the point representing the actual state of B and H in the magnet at the position in question) remain on the "outer loop" of the B/H curve (as shown in Figure 2), no permanent harm is done. When the disturbing mmf is removed, the magnet will return to its previous state. If the operating point moves down and to the left of the curve until it drops below the knee, as shown in Figure 3, the field of the magnet will not return to its original state once the disturbing mmf is removed. Instead, the operating point moves into the interior of the curve, along a line approximately parallel to the upper part of the outer loop. The magnetic field is therefore reduced and the magnet is partially demagnetized.
Figure 3: Partial de-magnetization of magnet
The external reluctance experienced by lines of flux from a magnet is a function of the permeability of air, which is very nearly that of free space, and the shape of the magnet, as well as the position of the line through the magnet. In 1920, Evershed (Ref. 1) derived a relationship from which an average B/H value could be found for a magnet as a function of its shape. Curves of B/H versus dimensional ratios for different shapes (such as round rods, bars of rectangular cross-section, round rods with central holes, etc.) based on this relationship have been widely published. Originally derived for use with highly permeable materials, the relationship works even better with those of low permeability with a suitable choice of effective pole spacing or length. The results agree well with measurements (based on magnetic moment) for intermediate magnet lengths but are less accurate elsewhere. Today most designs use magnets which are relatively thin in the direction of magnetization, compared to their cross-sectional dimensions and, as will be seen, the assumptions which go into the Evershed calculation are completely invalid for these shapes. The Evershed calculation also gives only one average value for each shape and cannot help in understanding local effects. A "hand calculation" to approximate the B/H of a permanent magnet bar as a function of length is described in Ref. 2. For these reasons, we decided to use the modern tool of finite-element-analysis on a range of dimensions, in order to obtain magnet design data not subject to the limitations of the old formula. We have compared the results to those of the Evershed formula and, where possible, to experimental measurements.
Evershed's calculation first modeled a long bar magnet as two poles of the same strength but opposite sign, one a source of magnetic flux and the other a sink, separated by a distance .
Each pole is surrounded by a sphere of radius . Each pole radiates or receives flux evenly in all directions, as shown in Figure 4.
Figure 4: 2-pole model of a rod magnet
The reluctance of a volume of space is:
= permeability of free space
= increment of path length of flux
A = cross-sectional area normal to flux path
For a sphere of radius r, the surface area is A =
Integrating from to infinity:
The total reluctance was then taken as twice this integral:
This assumption is an upper limit on and is approximately valid only for lengths much larger than .
The surface area for each pole of radius is:
Substituting into Equation (4):
Evershed then made the brilliant assumption that the magnet behavior of this model would resemble that of a real magnet which had an area at one of the two limbs (an end which could be described as a north or south magnetic pole) equal to that of one of the spheres.
where is the total area of the magnet.
The permeance is the reciprocal of reluctance , and so:
The permeance calculated here is the permeance of the space surrounding the magnet, not that of the magnet. Therefore:
However, the flux passing through the magnet is the same as that passing through the space around it, and so:
While B is highly variable from point to point in space, it is relatively uniform within the magnet (i.e. an average value is assumed) and so:
= average B in magnet
= Constant cross-sectional area of magnet
Since the closed-loop integral of must be zero in the absence of electric currents:
The negative sign means that the B/H curves of interest are in the second quadrant, where B is positive and H is negative, and can be dropped. Again, H is variable in space but is assumed to take on an average, constant value in the magnet, extending from one pole to another, and so:
It was observed from the plots of flux lines of a magnet of highly permeable material that the lines seemed to emanate from points somewhat inside the magnet ends, separated by a distance of about 70% of the magnet's actual length (as shown in Figure 5). The pole spacing was thus taken as:
Where is a function of magnet permeability.
Figure 5: Effective pole length from flux plots
Combining the above and solving for B/H:
= a constant depending on magnet material; 0.7 for steel or Alnico; 1.0 for ferrites, SamCo or NeoFe (low permeability materials)
= magnet length
= magnet constant cross-sectional area
= magnet surface area
This is Evershed's formula for the average permeance of a bar magnet of constant cross-sectional area. The surface area depends on the cross-sectional shape.
When these relationships are combined for round rods of length and diameter , using for area both the sides and ends of the bars, there finally results:
One result of the above is plotted in Figure 6, labeled "Evershed formula", as flux-density B (not B/H), derived from both the above formula and the B/H curve of a typical high-energy ferrite (M8B), shown in Figure 2.
Figure 6: Calculated and measured flux-density values
The results of computer finite-element calculations for average axial flux-density are also shown in that figure using the same material, the function plotted being the integral of the axial component times an increment of volume at that value over the entire magnet, divided by magnet volume:
In the same figure, average values of B are also plotted from measurements. The magnetic moment of the magnet sample was found, either from measurements by Helmholtz coil (Ref. 3), or by calculation from measurements of B versus position along the magnet axis. The average flux-density in the magnet was then found from:
III PRESENTATION OF RESULTS:
When the finite-element calculations are complete, one is faced with the question of how to present, in reasonably compact form, a useful summary of the voluminous results. It seems obvious to plot maximum and minimum values of B/H versus the length/diameter ratio and these are shown in Figures 7a and 7b. For an "average" value of B/H versus the ratio there are many possible choices.
Figure 7a & b: FEA results
We have chosen to use as an average, the B/H value from the magnetization curve corresponding to the axially-directed component of flux-density B which, if constant throughout the magnet, would produce the same magnetic moment as the magnet has from its actual state.
The magnetic moment of a magnet may be defined as a vector property of the magnet which produces torque T when the magnet is placed in a magnetic field B, according to the relationship:
It can be shown that the magnetic moment m is equal to:
= the axial vector component of the flux-density vector B
= an increment of volume of the magnet
At a distance from the magnet which is considered large compared to the magnet dimensions, the magnetic field is a function of the magnetic moment only. Only the axial component of the field is effective in producing moment, because the radial components cancel and the tangential component is zero.
For the model of a bar magnet consisting of two magnetic poles of opposite sign, of strength and spacing , the magnetic moment is:
but, for a real magnet, the flux through the moment is:
and if B is a constant, the average value across the cross-section is:
The minimum values of B/H, from the computer finite-element calculations, turn out to be surprisingly low, so that in many cases it appears impossible to avoid at least a small degree of self-demagnetization by shape. In that case, a magnet designer would want to know how much loss to expect; a very small loss due to shape magnetization would probably be preferable to the alternative of using a much larger amount of magnet material. Finding a way to present that information is a much more difficult task than for the other parameters, due to the wide variety of B/H curve shapes. Upon consideration it seems that a set of curves, each representing values of B/H (as a percentage of ) found in the corresponding amount of magnet (indicated as a percentage of the total volume) would serve this purpose. For example, for the magnet shown in Figure 8, which has an ratio of 0.25, it was found that 20% of the magnet volume had a B/H ratio equal to or less than 19% of , 70% of the volume had a B/H ratio equal to or less than 42% of , and so on.
Figure 8: Typical FEA plot
The complete results of the finite-element calculations of the (B/H)/ versus percent volume, as a function of ratio, are shown in Figure 9. The magnet material has a straight line B/H curve and permeability of 1.06 (which is typical for ferrites, neodymium-iron and some samarium-cobalt materials). These results must be considered somewhat tentative and used with care, because we were using a new program version with a number of "bugs" which had to be worked out. Since: 1) the average B values seem to be in good agreement with those derived from measurements, and 2) the calculated values of flux-density outside the magnet (when smoothed) agreed well with measurements, we felt therefore, that it would be reasonable to publish the results.
Calculations for the field of a partially shape-demagnetizing magnet based on these curves will necessarily be somewhat approximate, depending on the fraction of total material affected, shape of the B/H curve, etc. If a more accurate result is needed, a nonlinear finite-element model of the part would be required.
1. S. Evershed, IEEE Journal, Vol. 58, May 1920, pp. 780-837
2. R. Parker & R. Studders, "Permanent Magnets and Their Application", J. Wiley & Sons, 1962, pp. 172-173
3. S. R. Trout, "Use of Helmholtz Coils for Magnetic Measurements", IEEE Transactions on Magnetics, Vol. 24, No. 4, July 1988, p. 2108