Optimisation of design parameters to improve performance of a planar electromagnetic actuator

TL;DR

The paper addresses the limited actuation bandwidth of planar EM actuators by deriving closed-form expressions that relate design parameters to electromagnetic force and stiffness and by optimizing geometry to maximize force and stiffness per magnet volume. It develops a two-track planar coil model with a co-planar magnet, uses a first-harmonic representation to obtain explicit expressions for the fields, forces $F_x, F_z$ and stiffness $k_x, k_z$, and shows that the normalized performance scales as $F \sim 1/p^2$ and $k \sim 1/p^3$. The authors validate the theory experimentally with a MEMS force sensor in a feedback loop, achieving agreement within $<15\%$ across parameter sweeps and demonstrating that reducing pitch by factors improves force, while maintaining $z \ge p/4$ preserves harmonic behavior. The results indicate that micro-scale magnetic arrays fabricated with MEMS techniques can yield force and stiffness per unit volume two to three orders of magnitude higher, enabling high bandwidth levitation-based nano-positioners and high dynamic-range actuators.

Abstract

Planar electromagnetic actuators based on the principle of linear motors are widely employed for micro and nano positioning applications. These actuators usually employ a planar magnetic platform driven by a co-planar electromagnetic coil. While these actuators offer a large motion range and high positioning resolution, their actuation bandwidth is limited due to relatively small electromagnetic stiffness. We report optimization of the design parameters of the electromagnetic coil and the magnetic assembly to maximize the electromagnetic force and stiffness. Firstly, we derive closed-form expressions for the electromagnetic forces and stiffness, which enable us to express these quantities in terms of the design parameters of the actuator. Secondly, based on these derived expressions, we estimate the optimum values of the design parameters to maximize force and stiffness. Notably, for the optimum design parameters, the force and stiffness per unit volume can be increased by two and three orders of magnitude, respectively by reducing the pitch of the electromagnetic coil by a factor of 10. Lastly, we develop an electromagnetic actuator and evaluate its performance using a Microelectromechanical system (MEMS) based force sensor. By operating the force sensor in a feedback loop, we precisely measure the generated electromagnetic forces for different design parameters of the actuator. The experimental results obtained align closely with the analytical values, with an error of less than 15%.

Figures (6)

• Figure 1: Schematics showing (a) Isometric view of the electromagnetic actuator, (b) Top-view and (c) Side view showing only one track and the magnet for modeling purposes.
• Figure 2: Plots showing (a) the electromagnetic field ${b}_{z1}$ and its first harmonic ${B}_{z1}$ represented by the solid and the dotted lines respectively, (b) $\frac{\psi}{\tilde{l^2}}$ as a function of $\tilde{l}$, (c)$\frac{\phi}{\tilde{t}}$ as a function of $\tilde{t}$ and $\tilde{z}$ and (d) the electromagnetic X-force $\overline{F}_{x1}$ represented by the solid line and the numerically obtained X-force shown in the dotted red line. The magnetic field and the force are normalized with their respective maximum value $b_o =$ 0.39 $mT/A$ and $F_o =$ 0.58 $MN/A.m^3$ The values of $N$, $w$ and $p$ are considered 100, 100 $\mu m$ and 1 $mm$ respectively for calculation of $b_{z1}$. The values of $\tilde{l}$, $\tilde{t}$ and $\tilde{z}$ are considered 0.5, 0.25 and 0.25 respectively for the calculation of X- force $\overline{F}_{x1}$.
• Figure 3: Schematics showing: (a) Array 1 and (b) Array 2 designed entirely from permanent magnets, where the red and the black colour represents magnetic moment along positive and negative Z-axis respectively, (c) Array 3 designed using permanents magnets represented by red and non-magnetized material in between. The length of the magnets are considered as $p/2$, $p/\sqrt{2}$ and $0.64p$ for array 1, array 2 and array 3 respectively to provide maximum normalized force and stiffness.
• Figure 4: Schematic of the experimental set-up
• Figure 5: (a) Scanning Electron Microscope image of the force sensor with attached magnet, (b) Micro graphs showing two different magnets attached to the shuttle beams and (c) Image showing the two PCBs.