1. 1 pt. for doing the problem
  2. Getting within an order of magnitude is fine for all these questions, don’t worry about matching up exactly!
  3. 1 pt. for an area

Looking at the rectangular area (pad to pad in the horizontal, and the height of the motor in the vertical) I get about 8.64mm2. If you took the circular area, that’s fine.

  1. 1 pt. for a min capacitance, 1 pt. for a max capacitance, 1 pt. for the total displacement, 1 pt. for dC/dx. 4pts. total

The fingers are about 10cm long on my screen, just going over them with a ruler. Scale bar is about 1cm. So I’ll say they are 500µm long. Gaps are about 0.5mm on my screen. I’ll say they’re about 2µm wide. There are 6 gaps in total. The minimum overlap looks like 1.5cm on my screen. We’ll call that 75µm

Max overlap looks like the entire finger length, which is 500/75=6.7 times bigger than the minimum overlap. The capacitance is directly proportional to the overlap length

The total travel looks like the length of the fingers, so about 500µm.

The total change in capacitance over the length of travel is .

The change in capacitance of plates with respect to a changing overlap length is a constant.

  1. 1 pt. for the max capacitance, 1 pt. for dC/dx

The fingers look like they are about 5µm wide. The gap looks like it is about 2µm wide between the fingers when they fully overlap

This time we’ll find dC/dx mathematically. Assuming the fingers are parallel plates while passing by each other, the capacitance as a function of the changing overlap will be

Where t is the SOI thickness, and L is the nominal overlap length (0 in this case). Taking dC/dx we get

We did not take into account the fact that dC/dx will change in a complicated manner when the fingers don’t overlap, but this approximation is usually good enough.

  1. 1pt. for finding all the motors

Screenshots with the motors circled are shown below.

  1. 8 pts. total
  2. 1 pt. for an honest attempt

1 pt. for showing the 0.25 µm undercut

  1. 1 pt. for an honest attempt

1 pt. for showing the beam gets released

  1. 1 pt. for an honest attempt

1 pt. for showing that the perforated plate is not released.

For the figure below, the etch front is shown in blue. Your drawing does not need to be this detailed, only note that there is still oxide attached to the perforated plate.

  1. 1 pt. for an honest attempt

1 pt. for being roughly correct (within the right order of magnitude. It is fine if you did not account for every gap).

Find capacitance using the formula

There are three different gaps, the 1µm gaps in the problem description, , and the larger gap between these two gaps, . can be found by drawing out the geometry. We will say it is about 3µm wide. These capacitors all add together, so we get

  1. 5 pts. total.247 problems are meant to be open ended and usually do not have straightforward solutions. Did you make an honest attempt? Did you account for the two different areas with the different permittivities (oxide permittivity vs. air permittivity)? Did you find the new gap distance to achieve higher capacitance between structure 1 and structure 2?