Plastic Cup x(t), F(t) Moving Speaker Coil (24 AWG Copper) PM Material W g 2w hm Wm W 8w Pole Piece (Pure Iron Core) Cross-sectional geometry and dimensions of an axi-symmetric loudspeaker Assume that the coil has a steady current of i=1 A, flowing through enough turns of 24 AWG magnet wire to form a single layer coil within the air gap. Also assume that w = 2 cm and g = 2 mm, use FEMM or Maxwell 2D/3D to model the electromechanical characteristics of this axi-symmetric speaker core. Specifically: 1. If the PM material is NdFeB 52 MGOe with a fixed height of hm-4 cm, use FEMM or Maxwell 2D/3D to determine the appropriate magnet width, wm (in units of cm), such that the resulting air gap flux density is as close to 1.0 T as possible (+/- 1.0 cm). • Assume that the plastic cup has sidewall thickness of 1 mm and a relative permeability of μ-1 • In FEMM: use the "Point properties" tool to plot/measure the flux density magnitude in the center of the air gap. Don't simply estimate based on the flux colors. Make sure the "Output Window" is visible (in the View menu) Wm cm Upload an image of your resulting FEA simulation showing magnetic flux density in the air gap of your model (file size limit is 512 KB): Choose File No file chosen 2. These simulation results reveal a sub-optimal design characteristic of this speaker, specifically: ◇ Eddy currents in the iron core ◇ Hysteresis in the iron core O High permeability in the NdFeB O Flux saturation in the center core O Demagnetization in the NdFeB 3. This sub-optimal design characteristic could be corrected by: Increasing the center core width Using copper instead of iron for the core ◇ Reducing the NdFeB height ◇ Using more turns of magnet wire ◇ Using a smaller gage magnet wire 4. Using FEA, determine the magnitude of force (in N) exerted on the middle of the moving coil • In FEMM: select the coil using the "Areas" tool and compute the Lorentz force block integral • Do not include vector notation (unit vectors, etc.) F= N Upload an image of your resulting FEA simulation showing the Lorentz force exerted on the coil of your model (file size is limit 512 KB): Choose File No file chosen 5. Now analytically calculate the force magnitude exerted on the middle of the moving coil (using Lorentz's law). Assume that the air gap flux density is exactly 1.0 T. Hint: first enter the total number of active turns in the magnetic field (N) used to compute the Lorentz force: N= The resulting Lorentz force magnitude is thus: F= N