Homework #1

Assigned: Jan. 13, 2004
Due: Jan. 20, 2004
Points: 30

1. Points: 15
   Consider a beam of length L, width w, and thickness t, that is clamped on both logitudinal ends. Imagine a plane below the beam separated by a gap g₀. Assume that a voltage V is applied between the beam and the plane. This creates an electrostatic force between the two. If the plane is held fixed, the beam will elastically deform under this force.
   For now, assume that the electrostatic force acting per unit length of the beam is given by (e ₀wV²)/(2g²) where e₀ is the permittivity of free space.
   • Check the dimensionality of the above formula for the electrostatic force by using the following fundamental dimensions: M (mass), L (length), T (time), and C (Coulomb for charge).
   Fix the proportions of the beam and gap as: w = L/20, t = g = L/50. Let d be the maximum deflection of the beam under the force. Assume silicon to be the material with an Young's modulus of 150 GPa.
   • For V = 5 V, plot d/L for different sizes (i.e., vary L over a wide range from 10 m to 100nm. Use log-scale.
   • For d/L = 0.02, plot the required voltage vs. the length. Again, vary the length over a wide range as before.
   Do your results indicate why electrostatics is the preferred mode of actuation in MEMS but not in the macro world?

2. Points: 5
   Use dimensional analysis to see how the resistive power loss per unit volume scales with size. Resistive power loss is the loss of heat due to Joule heating in an electrical resistor. We are interested in resistive loss per unit volume because this will be of interest if a device has its own power. What does your analysis imply for a self-powered device that is scaled down in size?

3. Points: 10
   Search the Internet to identify a commercially available product that uses a MEMS component. All the devices noted in the Power Point presentation shown in the first class cannot be used for this purpose. It means that you need to really "search" the Internet.
   For the product that you identify, give a very brief description of the MEMS component it has, the working principle of the component (as much as you can understand), and what process was used to make that MEMS component. Please do not forget to the cite the URL of the source.