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):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "http://localhost:6011/en.wikipedia.org/v1/":): {\displaystyle (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 $(BH) max$ can be illustrated by considering a simple magnetic circuit containing a permanent magnet of volume $Vol mag$ and an air gap of volume $Vol gap$ . The total magnetic energy in the gap (volume-integrated magnetic energy density) is then directly related to the volume-integrated $- BH$ in the magnet:^[5]

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

thus in order to achieve a particular magnetic field $B gap$ in the gap, the required volume of magnet can be minimized by maximizing $- BH$ in the magnet (by choosing a material with high $(BH) max$ , and operating it at the $(BH) max$ point by matching the magnet geometry to the circuit's reluctance).

References

^ "What is Maximum Energy Product / BHmax and How Does It Correspond to Magnet Grade? | Dura Magnetics USA". 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.{{cite book}}: CS1 maint: multiple names: authors list (link)

[1] "What is Maximum Energy Product / BHmax and How Does It Correspond to Magnet Grade? | Dura Magnetics USA". 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.{{cite book}}: CS1 maint: multiple names: authors list (link)

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