EE215 Homework 2 Solutions

Transcription

1 EE15 Homewor Solutions 1 ayout a comb drive resonator in the PolyMUMPS process using -Edit in MEMS Pro as shown in the figure below. Use the poly 1 layer (h µm for fabrication of the comb drives and folded springs. You can construct the resonator using elements from the MEMS Pro cell library. The length of the folded springs will be 150 µm, the width of the beams will be W µm wide and they will be separated by 18 µm. The comb drive fingers will be 40 µm long and µm wide, with a µm gap between the fingers. The fied and released fingers should have an un-deflected overlap of 0 µm. You should not need to use poly in your layout ecept for the bond pads. Be sure to use bond pads from the MEMS Pro cell library! Save the final layout file as Your- Name_Resonator.tdb and mail it to the instructor. Mae a solid model of your resonator. Eperiment with the scaling in the z-direction to mae the solid model easier to visualize. µm lines and spaces Resonator 18 µm µm 150 µm 40 µm 0 µm igure.0. Comb drive resonator. The things that I usually find missing are the dimples and etch holes. If you do not include dimples then the released parts will tend to get stuc to the substrate, particularly the shuttle plate that is suspended at the end of the fleible folded springs. If the lateral etch holes in POY1 are not included as described in the PolyMUMPS process at a spacing of <0 µm then the parts will not be released!

2 EE15 Homewor Solutions Calculate the resonant frequency for the comb drive resonator in problem 1 above. a. Brea the folded spring down into parts. irst, consider the two beams, each of length, shown in the figure below. Each of these beams is a fied-guided beam, and they are connected in series so that the spring constant is given by 1/ total 1/ 1 +1/ : Two fied-guided beams of length connected in series K effective 1/ 1 +1/ where 1 1EI/ so that effective 6EI/ Net, consider the two springs shown below that are connected in parallel. The effective spring constant for springs connected in parallel is given by total 1 + :

3 EE15 Homewor Solutions Two springs connected in parallel K effective 1 + where 1 6EI/ so that effective 1EI/ inally, consider the two springs shown below that are connected in parallel: Two springs connected in parallel K effective 1 + where 1 1EI/ so that effective 4EI/ So that the total effective spring constant for the two folded springs that are connected in parallel is given by: ( where I ( 1 4 Eh W sys EI hw 1 The spring constant for the system is then given by: Spring Constant Young's Modulus (E Beam Thicness (h Beam Width (W Beam ength ( 1.60E+11.00E-06.00E E-04

4 EE15 Homewor Solutions [N/m] 1.5E+00 b. Use the effective spring constant sys found above to estimate the resonant frequency of the comb drive resonator using the following formula: f r 1 π M p sys M 1/ Where M P and M are the masses of the shuttle plate and of the supporting beams, respectively. This approimate epression is found using the Rayleigh Ritz energy method. The effective mass can be found from summing each of the individual components of the resonator:

5 EE15 Homewor Solutions Plate Element ength (um Width (um Thicness (um olume (um^ Total (um^ Yellow Blue Cyan Green ingers Total (um^ 80 Total (m^.8e-14 Density (g/m^ 0 Mass Plate (g 6.60E-11 Beams Mass Beams (g 1.1E-11 Effective Mass (g 7.01E-11 The resonant frequency is then given by: Resonant requency 1.5E+00 Effective mass 7.01E-11 f [Hz],407 Assume that silicon will fracture when the aial stress reaches 1 GPa. ind the maimum length of a vertical silicon rod which, under the action of its own gravitational load, will not eceed this fracture stress. Assume the density of silicon is ρ 1 g/m and that the acceleration due to gravity is g 9.8 m/sec. Are you surpised? σ racture 1 GPa, ρ 1 g/m σ /A mg/a (ρ(π/4d g/ (π/4d ρg σ racture /ρg (1 9 N/m /(1 g/m (9.8 m/sec meteres 4 Model a released mirror for a MEMS abry-perot interferometer in the SOI-MUMPS process as a stiff plate (mm mm fabricated in the device layer of the µm thic SOI wafer suspended by four diagonal fied-guided cantilever beams (h µm thic, W 0 µm wide, long at each corner of the mirror as shown in the figure below. ind an epression for the deflection δ of the mirror as a function of the support arm length assuming a force is acting vertically on the mirror. What is the longest length for the support arms that you can fit into the allowed 8mm 8mm die site including the central mm mm mirror? You can assume Young s modulus for silicon is E 169 GPa parallel to the wafer flat for a (0 wafer, GPa at 45 degrees to the wafer flat and that Poisson s ratio is ν 0.8. or a single fied-guided beam, we will use equation [.19] from the tetboo:

6 EE15 Homewor Solutions y( 1EI ( l [.19] l The maimum deflection δ 1 for a single beam would then be: δ 1 1EI 1EI EI 1 ( If we put this into the form of Hoo s law, δ, we see that the spring constant 1 for one beam is: 1EI 1EI 1 δ 1 1 And the spring constant 4 for four beams connected in parallel is four times stiffer: 4 48EI So the deflection δ 4 with four springs in parallel would be: δ 4 EI 48 The longest length of cantilever beams that will fit within the 8mm 8mm die site including the mm mm mirror would be: mm

7 EE15 Homewor Solutions MEMS Pro solid model of mm mm mirror supported by four 4.5 mm fied-guided beams 5 Calculate the voltage that will be required to deflect the mirror for the abry-perot interferometer you designed in problem 4 using an electrostatic actuator. You can assume that the area of the parallel plate capacitor is the same as the mm mm mirror surface. Assume the initial gap from a second bonded wafer with a counter-electrode is.4 µm and that you need to be able to deflect the mirror surface until the final gap is / of the initial value, or 1.6 µm. irst, we will calculate the required force to deflect the mirror by the required 0.8 µm. rom problem 4 we have the deflection of the four springs to be: δ 4 EI 48 Solving for and substituting δ m: 48EI δ ( Pa ( 4.5 m N 0.54µ N ( 0 m( m ( 0.8 m 1 We can find the voltage required to generate this force from formula [.1] for the electrostatic actuator:

8 EE15 Homewor Solutions e C ε 0 A z ( g z 0 ε A 0 e ( g z [.1] (.4 m 0.8 m ( 0 m 0.54 N 6 irst, find the capacitance C 0 for the different beam lengths: C C C C A ε d ( ( (00 A ( or the cantilever beam, the spring constant can be calculated as:

9 ( / y( 4EI EE15 Homewor Solutions 8EI 8EI y( 8EI 8E wt m N / m 1.N / m 0.68N / m ( ( ( (1.5(1 9 ( ( or the fied-fied cantilever beam, the spring constant can be calculated as:

10 EE15 Homewor Solutions m N m N m N EI EI EI y m (1.5( ( /.8 / 64.0 / ( (0 84(