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ELEC 315 - Assignment 4

Due Dec. 7, 2023

Winter 2023-2024 Term 1

1.   We want to build a micro-positioning system that produces a maximum travel of 5 μm by using a piezoelectric linear actuator beam with a square cross section (a2) and  length (l) as shown below. A piezoelectric material with piezoelectric coefficients d13 = -3.0x10-8, d33 = 4.6x10-8, d24 = 3.2 x10-8  and d25 = 3.0 x10-8  cm/V ( Si = djiEj ) and a dielectric strength of 8 x105  V/cm is used.

(a) Write the matrix of piezoelectric coefficients and explain which directions can be used to apply electric fields for the linear actuation in the desired direction (3).

(b) For a maximum power supply voltage of 110 V and a safety parameter of 2, find the minimum dimensions of the beam (a and l). Explain which direction for electric field is the most effective for creating the required linear travel and how we should place electrodes to apply such field to the actuator beam.

2.   We have a transparent film of BaTiO3  with a side of l = 100 µm and thickness t = 0.1 µm, as shown below.  Under Curie temperature, this ferroelectric material is poled in direction 3 and shows an anisotropic dielectric constant that is εr(!) = 3800 perpendicular to polar axis and εr(!) = 150  parallel to polar axis. Assume that the electronic polarizability of each atom is 2x10-36  Fcm2  and lattice parameter is approximately a = 0.4 nm (assume the unit cell is almost cubic, axax1.09a).

(a) Calculate the refractive index and speed of visible light in this film. Explain the assumptions and approximations you make for this calculation with respect to real BaTiO3 crystal structure.

(b) If we place two electrodes on top and bottom of the film (colored area), estimate the capacitance at 100kHz.  Find the loss tangent at this frequency knowing that  the time constant associated with the polarization mechanism is 0.8 µs. Explain   the model you use for frequency dependence of the dielectric constant.

3. Consider two materials Bi and Al with Xm = -20 x 10-5 and Xm = -2 x 10-5 respectively. Find the magnetization vector M and the magnetic field B in these materials when they are under an external magnetic field of 1 T in direction x. What are the types of these two magnetic materials?

A soft ferromagnetic material with Xm = 100 and a coercive field of u0Hc = 10-5 T is used for information storage. If a solenoid with 100 turn per unit length is used as the writing head, what is the minimum current required that we need to pass through the solenoid to write a data.