Surface magnetic field strength is an important technical indicator of finished magnetic steel products. As the name suggests, it refers to the magnetic induction intensity of the surface. In the industry, it’s sometimes colloquially referred to as the surface magnetic field strength, or surface field. For applications requiring the use of spatial magnetic fields, surface magnetic field strength, or the magnetic induction intensity value at a specific point, is usually considered a crucial technical requirement. It’s important to clarify that surface magnetic field strength has a direction; we typically refer to the surface magnetic field value perpendicular to the magnetic pole face.
Can surface magnetic field strength be calculated?
If so, how?
This is a question of great concern. Some calculation formulas can be found online, and many magnetic material manufacturers’ websites also offer corresponding calculation programs for convenience. However, those who have used these tools will find that they can only calculate the surface magnetic field strength of cylindrical and rectangular magnets, and specifically the surface magnetic field strength at the center. Is it difficult to calculate for other shapes? Is it also difficult to calculate the maximum surface magnetic field strength? Yes, accurately calculating the surface magnetism is indeed very difficult and complex. For cylindrical and rectangular magnets, we assume that the magnetic field distribution is ideal and symmetrical, the center surface magnetism is ideally perpendicular to the magnetic pole face, and the permeability in the air gap is equal to the permeability of the magnet, both equal to 1.0. Based on these conditions, a relatively simple calculation method can be found, which can be found in TDK’s calculation formula:
The above formula can be used to create an Excel calculator for convenient calculations, which is also the source of formulas for many website calculation programs. However, in actual calculations, it’s found that the accuracy of the results seems insufficient. For some products, the calculated value differs significantly from the measured value. Take an N50 D10*10mm magnet as an example. Using a gaussmeter placed against the center of the magnetic pole surface, assuming the probe is tightly attached to the magnet with no air gap and excluding instrument errors, and taking Br as 14kGs according to the formula, the center surface magnetic field should be 6261Gs. However, in reality, no matter how you measure, you cannot reach this value, not even with an N54 magnet. What’s the problem? Is TDK’s formula inaccurate?
To analyze this problem, we need to understand the benchmark upon which the formula is based; we cannot simply adopt it blindly. First, in the formula, X represents the distance from the calculation point to the magnet surface. Although we measure directly with the probe against the magnet surface, the gaussmeter probe itself doesn’t expose the Hall chip; it has a protective shell. For example, the KANETEC gaussmeter from Japan, widely used in the industry, has a Hall chip with a transparent protective shell of approximately 0.2mm. Furthermore, with repeated measurements, you’ll notice a difference in surface magnetic field between the black sample and the finished product. This is due to the coating on the finished product. For example, a nickel-copper-nickel coating typically has a thickness of 0.02mm on one side. Since nickel itself is a magnetically conductive material, it shields the magnetic field, thus reducing the measured value. Moreover, as we’ve emphasized, the surface magnetic field measured by a gaussmeter isn’t an ideal point, but rather a small area. The actual magnet orientation and uniformity cannot be perfect. Therefore, based on these limitations and extensive practical experience, an additional compensation of 0.4-0.5mm for X is recommended. For Pc values below 3, compensate 0.5mm; for Pc values above 3, compensate 0.4mm.
The above explains the compensation for the X value. An even more important point is that the formula holds true only if the permeability of both the magnet and the air gap is equal to 1.0. We know that the permeability of air is 1.000065, very close to 1.0, but the permeability of magnets is not so ideal. In fact, reaching 1.02 is already a very good value. Most N-series and M-series neodymium iron boron magnets exceed 1.05, some even reaching 1.1. Why does permeability affect the test data? We still need to return to the demagnetization curve for analysis and understanding:
Br remanence and Bdi intrinsic magnetic flux
As can be seen from the demagnetization curve, the actual remanence of a magnet in an open-circuit state is not the ideal Br value, but rather lower than Br. This is called the intrinsic magnetic flux density Bdi. The reason for this is that the initial state of the blue J-H demagnetization curve is not ideally parallel to the X-axis, but rather inclined. In this case, the Br value will definitely be greater than Hcb, and the restoring permeability μrec = Br/Hcb cannot be 1.0. Another situation is that the B-H line has an inflection point, and the magnet’s operating point is below the inflection point. In this case, the actual Bdi will be much lower than Br, causing the calculated result to deviate significantly from the actual value, as shown in the figure below. This also explains why the actual center surface magnetism of the N54 20*10*1mm magnet is not only far lower than the theoretical calculated value, but also very unstable.
Combining the above, modifying Br in the formula to Bdi, with appropriate X compensation, will result in a more accurate calculation formula. However, in practice, Bdi is also difficult to calculate. The best method is to calculate it from the graph based on the actual demagnetization curve and Pc value, but this is also quite cumbersome. It is recommended to calculate it using the following approach:
1. First, the operating point must be above the inflection point of the B-H demagnetization curve with an appropriate margin. This is especially important for N-materials with a magnetic energy product of 45 or higher and thin sheets with Pc less than 0.6. If it is below the inflection point, it means the magnetism is unstable. Even if Bdi is calculated from the curve to determine the center surface magnetic fluctuation, it will be large. Hcj must be increased to maintain the operating point below the inflection point or to prevent the BH line from showing an inflection point.
2. Assuming the first point is satisfied, Bdi = Br∙(Pc+1)/(μrec+Pc), where μrec is determined based on actual values. Generally, it is 1.08-1.1 for N40 and below, 1.06-1.08 for N40 and above, 1.05-1.06 for M-grade, 1.04 for H-grade, and 1.03 for others.
The final formulas for calculating the surface magnetism of the center of cylindrical (or nearly cylindrical) and rectangular (or nearly rectangular) magnets are as follows:
Note: The above formulas are only applicable to permanent magnets with linear demagnetization curves, such as NdFeB, SmCo, and ferrite. They are not suitable for permanent magnets with nonlinear demagnetization curves, such as AlNiCo, FCrCo, and various soft magnetic materials. Additionally, this method is also not applicable to magnets with oblique orientations whose magnetization direction is not perpendicular to the magnetic pole plane.