Maximum energy product

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]

E_{\rm {gap}}={\frac {1}{2\mu _{0}}}(B_{\rm {gap}})^{2}{\rm {Vol}}_{\rm {gap}}=-{\frac {1}{2}}B_{\rm {mag}}H_{\rm {mag}}{\rm {Vol}}_{\rm {mag}}=-E_{\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 to achieve a target flux density in the air gap is minimized. This expression assumes that the permeability in the core that is connecting the magnetic material to the air gap is infinite, so unlike the equation might imply, you cannot get arbitrarily large flux density in the air gap by decreasing the gap distance. A real core will eventually saturate.

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.
^ Umans, Stephen D. (2014). "1.5 Permanent Magnets". Fitzgerald & Kingsley's Electric Machinery (7th ed.). McGraw-Hill. p. 33. ISBN 978-0-07-338046-9.

[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] Umans, Stephen D. (2014). "1.5 Permanent Magnets". Fitzgerald & Kingsley's Electric Machinery (7th ed.). McGraw-Hill. p. 33. ISBN 978-0-07-338046-9.

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Definition and significance

See also

References