Maximum energy product

Historical trends in the maximum energy product of permanent magnets (MGOe units).

In magnetics, the maximum energy product is an important figure-of-merit for the strength of a permanent magnet material. It is often denoted $(BH) max$ and is typically given in units of either $kJ/m 3$ (kilojoules per cubic meter, in SI electromagnetism) or $MGOe$ (mega-gauss-oersted, in gaussian electromagnetism).^[1]^[2] 1 MGOe is equivalent to 7.958 kJ/m³.^[3]

During the 20th century, the maximum energy product of commercially available magnetic materials rose from around 1 MGOe (e.g. in KS Steel) to over 50 MGOe (in neodymium magnets).^[4] Other important permanent magnet properties include the remanence ( $B r$ ) and coercivity ( $H c$ ); these quantities are also determined from the saturation loop and are related to the maximum energy product, though not directly.

Definition and significance[]

(BH) max

can be graphically defined as the area of the largest rectangle that can drawn in the second quadrant of the B-H loop.

The maximum energy product is defined based on the magnetic hysteresis saturation loop ( $B$ - $H$ curve), in the demagnetizing portion where the $B$ and $H$ fields are in opposition. It is defined as the maximal value of the product of $B$ and $H$ along this curve (actually, the maximum of the negative of the product, $- BH$ , since they have opposing signs):

(BH)_{\rm {max}}\equiv \operatorname {max} (-B\cdot H).

Equivalently, it can be graphically defined as the area of the largest rectangle that can be drawn between the origin and the saturation demagnetization B-H curve (see figure).

The significance of $(BH) max$ is that the volume of magnet necessary for any given application tends to be inversely proportional to $(BH) max$ . This is illustrated by considering a simple magnetic circuit containing a permanent magnet of volume $Vol mag$ and an air gap of volume $Vol gap$ , connected to each other by a magnetic core. Suppose the goal is to reach a certain field strength $B gap$ in the gap. In such a situation, the total magnetic energy in the gap (volume-integrated magnetic energy density) is directly equal to half the volume-integrated $- BH$ in the magnet:^[5]

{\frac {1}{2}}(B_{\rm {gap}})^{2}{\rm {Vol}}_{\rm {gap}}=-{\frac {1}{2}}\mu _{0}B_{\rm {mag}}H_{\rm {mag}}{\rm {Vol}}_{\rm {mag}},

thus in order to achieve the desired magnetic field in the gap, the required volume of magnet can be minimized by maximizing $- BH$ in the magnet. By choosing a magnetic material with a high $(BH) max$ , and also choosing the aspect ratio of the magnet so that its $- BH$ is equal to $(BH) max$ , the required volume of magnet is minimized.

References[]

^ "What is Maximum Energy Product / BHmax and How Does It Correspond to Magnet Grade? | Dura Magnetics USA". 15 September 2014. Retrieved 2020-01-20.
^ "Glossary of Magnet Terminology". K&J Magnetics. Retrieved 2021-01-31.
^ eFunda: Glossary: Units: Energy Density Units: Megagauss-Oersted (MG⋅Oe)
^ "COBALT: Essential to High Performance Magnetics" (PDF). Arnold Magnetic Technologies. 2012.
^ Fitzgerald, A.E.; Kingsley, Charles, Jr.; Umans, Stephen D. (2003). Electric Machinery (6th ed.). McGraw-Hill. p. 34-. ISBN 978-0-07-366009-7.

[1] "What is Maximum Energy Product / BHmax and How Does It Correspond to Magnet Grade? | Dura Magnetics USA". 15 September 2014. Retrieved 2020-01-20.

[2] "Glossary of Magnet Terminology". K&J Magnetics. Retrieved 2021-01-31.

[3] Funda: Glossary: Units: Energy Density Units: Megagauss-Oersted (MG⋅Oe)

[4] "COBALT: Essential to High Performance Magnetics" (PDF). Arnold Magnetic Technologies. 2012.

[fitzgerald-5] Fitzgerald, A.E.; Kingsley, Charles, Jr.; Umans, Stephen D. (2003). Electric Machinery (6th ed.). McGraw-Hill. p. 34-. ISBN 978-0-07-366009-7.

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