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** Magnetic Force Equations based on Computer Simulation
and the Effect of Load Line **

** Christina Chen, Magnetics Specialist for Quadrant Inc. and Magnet Energy LLC
**

**Abstract**

The Maxwell magnetic force equation F = B^2A/(2u_0) can be used for determining the magnetic force of magnetic components, where F is the force in newton (N), B is the flux density in tesla (T), A is the area of cross-section in square meter (m^2), and u_0 is the permeability of the vacuum (4pi*10^-7 H/m). The formula can be converted to an easy to remember expression of F = 40B^2A, in which the unit of A is cm^2. This equation says that if the field is 1T, and the area is 1cm^2, then the magnetic force is 40N or 4kgf. However, it is somehow difficult to determine the B value in many practical cases, and the accuracy is usually not satisfactory. Computer simulation using finite element method can determine the magnetic forces with various boundary conditions, but usually it is not convenient for industrial users. In this presentation, we report several simple equations, which are established based on the large database generated by using 3D computer simulation. The users can use the equations to obtain the force by simply inputting the magnet's B_r, area and thickness. The effect of load line is also analyzed in this talk.

Infolytica's MagNet software was chosen for the FEA simulation. Parameterization function with newton tolerance 0.1% was used to systematically solve the problems for NdFeB cylinders, rings, and rectangular blocks interacting with CR1010 steel. The result database for each gap in a single boundary condition includes 62500 data points for rectangular blocks. The gaps between the magnets and steel plates are in the range of 0.01 - 15mm with 23 unequal intervals. The itemized data were then plotted and analyzed to establish the force equations for the magnets with relative high load lines: F = B_r^2(aA^2+bA).

In the equation, "a" and "b" are the function of magnets' thickness, which vary for the cylinders, rings, and blocks. The effect of boundary condition is tremendous. The magnets with lower load lines have significant lower magnetic forces, and their magnetic force values vs the area cannot generate satisfactory equations. Details for all the magnet shapes with different boundary conditions and the effect of load line will be analyzed.

**Biography**

Christina Chen obtained her PhD in Materials Engineering from University of Dayton (UD) in 1993. Dr. Chen has many years' experiences in both industrial and academic areas in the field of magnetic materials and their applications. Her experiences also include magnetic circuit designs & computer modeling for electromagnetic devices. She is a Magnetics Specialist for Quadrant Inc. and Magnet Energy LLC.
Prior moving to the Bay Area, Dr. Chen had worked in academic area for 11 years and in the industrial field for more than a decade, including EEC & GE Global Research. She worked in and then led the UD magnetics Lab; where the Lab founder discovered the modern rare earth permanent magnetic materials in middle 1960's. Dr. Chen was the Principal Investigator for many research programs founded by NASA, NSF, DOD, DOE and others in the past, which resulted in patented commercial products, as well as 90+ publications in various professional journals and conference proceedings.

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Last updated on 03/22/2015