A magnetic circuit refers to a physical system comprised of components designed to generate and confine magnetic flux. Typically, this flux is generated by either permanent magnets or coils, with the flux paths being constructed from high-permeability ferromagnetic materials, such as iron. In Figure 1, we observe an illustration of a magnetic circuit [1] that includes two iron components with a specific cross-sectional area $\mathrm{S}=4c{\mathrm{m2}}^{}$ . One part is stationary and the other is moving; they are separated by two air gaps of length $\mathrm{L.\; N}$-turn coil wrapped around the stationary part, carries the current I ( $N=300;I=1A$) and is responsible for the flux in the circuit.
Force on the moving part can be determined by calculating the change in the magnetic energy that would be produced by moving the part over a small distance . Force magnitude is obtained as:
This simple method is based upon the virtual displacement principle and it is often used to calculate forces in magnetic devices.
Since the permeability of iron is much greater than the permeability of air (), the magnetic field in the iron (), as well as leakage flux, are neglected. Ampere’s law for the contour in Figure 1. can be written as:
where ${\mathrm{Ha}}_{}$ represents the magnetic field in the air gap.
With no leakage and no field in the iron (), the entire magnetic energy of the system is stored in the volume $V=SL$ of the air gap and can be computed as:
Assuming uniform field distribution in the air gap allows to calculate $W$ by simply multiplying energy density and volume $V$. The circuit contains 2 air gaps, therefore, after doubling the energy in eq.3 and substituting eq.2 in eq.3:
(eq. 4)
For two different air gap lengths${\mathrm{L1}}_{}=1.5mm$ and ${\mathrm{L2}}_{}=2.5mm$ , the change in magnetic energy can be computed as:
(eq. 5)
Equation 5, combined with eq. 1, yields the following expression for force magnitude:
(eq. 6)
To accurately represent the situation in the analytical example, where force has been computed using energies for $1.5mm$ and $2.5mm$ air gaps, the CAD model of the circuit has been created with two air gaps of different lengths:${\mathrm{L1}}_{}=1.5mm$ and ${\mathrm{L2}}_{}=2.5mm$ (Figure 2.).
To exploit the problem's inherent symmetry, it suffices to simulate just one-half of the space, corresponding to half of the magnetic circuit. The simulation is conducted using the EMS Magnetostatic study, with a symmetry factor set at 0.5.
Current current-driven Wound Coil with $300$ turns and $1A/turn$ is added to the coil region. Its Entry and Exit Ports are faces in the plane of symmetry.
The following instructions outline the process for creating a new material library, defining a custom material, and applying that material to an element within your model:
Copper is assigned to the coil region, and air to the surrounding volume.
The normal component of the flux density at the plane of symmetry ( $xy$ plane in Figure 2.) is zero (all flux lines are parallel to this plane). Therefore, the Tangential Flux boundary condition should be applied on all surfaces that belong to the plane, including air and coil cross-sections. To do so:
To add a coil to a Magnetostatic study:
Figure 4 - Entry and exit ports of the coil
EMS automatically calculates the distribution of nodal forces without requiring any user input. However, when it comes to calculating forces or torques on a rigid body, it is necessary to define the specific part on which the force or torque should be computed before initiating the simulation.
The quality of the mesh in the air gap region is of critical importance for accurate force calculation. EMS allows user to take full control over the mesh resolution.
To mesh the model:
Right-click Study icon and select Run , to execute the simulation. Once the computation is done, the program creates five folders in the EMS manager tree. These folders are Report, Magnetic Flux Density, Magnetic Field Intensity, Current Density, and Force Distribution.
It is a good habit to first view the magnetic flux density in the model, including the outer air. This action indicates whether the outer air boundary is far enough.
By examining Figure 5. it becomes clear that the magnetic flux density is very small on the outer air boundary. Thus, the air box is large enough. Had it been otherwise, the air box surrounding the magnetic circuit would have to be larger.
In the EMS Manager tree, Double-click Result Table to display the force result.
Remember that, because of the symmetry about the xy plane, only half of the problem has been modeled. Thus, $\mathrm{Fx}$ and components must be multiplied by a factor of 2, while the $\mathrm{Fz}$ component cancels out. Since $\mathrm{Fy}$ is very small compared to $\mathrm{Fx}$ , the resultant force is virtually in the X direction with a magnitude of $\mathrm{FxTotal}=2x1.507=3.014N$ . The analytical solution compares very well with the EMS result.
Analytical Virtual Work Solution | EMS Result | |
Force [N] | 3.02 | 3.014 |
The application note delves into the physics and simulation of magnetic circuits, focusing on force computation within a specific configuration. Through analytical derivations and simulation in Solidworks EMS Magnetostatic study, the note demonstrates the calculation of forces using principles like Ampere's law and virtual displacement. Key components such as iron cores, air gaps, and coils are meticulously modeled, with materials assigned and boundary conditions set accordingly. The simulation results, including magnetic flux density and force distribution, are compared with analytical solutions, showcasing close agreement and validating the simulation approach. By accurately capturing the behavior of magnetic circuits, the note offers valuable insights for engineers working in electromagnetics and device design. Additionally, it provides practical guidance on modeling techniques and simulation setup.
[1] Electromagnetics and calculation of fields, by Nathan Ida and Joao P. A. Bastos, 2nd Edition, page 183-184. Publisher: Springer-Verlag;
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