Design of micro accelerometer (стр. 2 из 3)

This error is much bigger and it will be dominant for accelerometer resolution.

Maximum acceleration

Polysilicon, which is piezoresistor’s material, can survive only if applied strain is less then 1%. If we use the same equation which was used to find strain four sections earlier we obtain the maximum allowable strain is equal to

Found acceleration is very huge. But for 100g acceleration deflection is already of such magnitude, that small deflection assumption is hardly valid. For larger then 100g acceleration large deflection analysis must be used. At large deflection elongation of beams can’t be neglected and it will affect resulting strain. Therefore, maximum acceleration found above shouldn’t be considered as true value. But from earlier analysis we can conclude that designed sensor satisfies original spec to be able to measure acceleration in range -100g~100g.

Dynamic model analysis

Etching time

For further analysis we need to know depth of cavity under seismic mass. In order to find it we have to find etching time first. To etch the silicon

process is used. To release proof mass etching time should be enough to etch the longest distance of silicon covered by proof mass. According to chosen design the maximum length is , where is distance between etching holes. So, the minimum etching time is

Assuming that etching time will be 32.5 min resulting, cavity depth is

Coefficients of basic equations

In order to predict behavior of the device under dynamic acceleration, dynamic model has to be constructed. Basic equations governing this model are following:

Where mass and specific spring constants were found in static model analysis:

Other coefficients have to be found.

and are moments of inertia around axes X and Y respectively. Because of symmetric design of proof mass these moments are equal to each other.

Ratio of total area of etching holes to area of proof mass is only about 0.6%, therefore, influence of holes on moment of inertia is neglected. The different density of materials added during MOSIS process is also neglected. So, to calculate moment of inertia we will use the same equation as for solid box.

Where a and b are dimensions of box in plane which is perpendicular to axis of rotation. To calculate it, it is needed to calculate thickness of proof mass first. Thickness of added alumina layer is

Then the total thickness of proof mass is

Now, moment of inertia can be calculated

Next step is to find damping constants. For normal motion only they can be found from damping force

Whose solution

is known from linearized Reynolds equation. Solution with subscript “0” represents action of gas between moving plates when frequency of motion is low (small squeeze number). In that case it acts as pure damper. At higher frequencies solution “1” becomes dominant and gas film acts as spring. Such behavior of film is not desirable. Therefore, accelerometer should be used under acceleration whose frequency is less then certain value. This so called cut-off frequency will be estimated later. Now, only solution F₀ will be considered.

Damping force can be approximated by neglecting the у term in series solution as follows

Where it is used that moving plate has square shape and constant 0.42 is correction coefficient due to its unit aspect ratio.

Finally, the damping constant of normal motion is

For tilt motion expression of angular momentum is also known in form of series solution. According to frequency of acceleration it can act as damper or spring. And we again consider only damping behavior.

In equation above it is applied that aspect ratio is unit. Now, substituting expression for у and treating

as angular velocity, we can obtain damping of tilt motion

The series converges rather fast, therefore only first term will be calculated for tilt motion damping estimation. Also last term in denominator will be neglected.

Damping coefficients of tilt motion around X and Y axes are equal because of symmetry of proof mass.

Now, all nine coefficients of basic equations are know and system of differential equations can be solved.

Natural frequencies

For normal motion natural frequency is

Natural frequencies of rotation around X and Y axes are again the same because of symmetry of proof mass:

Damping ratios

From damping coefficients we can calculate damping ratios for normal motion

and for tilt motion

Where subscript

represents that tilt for tilt motion does not matter which axis we will choose for calculation.

Cut-off frequencies and squeeze numbers

Using one term approximation in series solution we can get value of cut-off squeeze number

It is applied in above equation that aspect ratio в is equal to one. Next we can approximate cut-off frequency

And for the tilt motion:

Because the main purpose of gas film is to provide damping of the device, spring behavior must be avoided. To satisfy this spec operation frequency should be lower then cut-off frequency.

As we can see cut-off frequency is much higher then natural frequency (three orders of magnitude higher). And because useful bandwidth is usually of order of natural frequency we can suppose that in designed accelerometer gas film will behave as damper always.

Sensor system simulation

Equivalent circuits

Equivalent circuit of normal motion is presented in Figure 4.

Figure 4. Equivalent circuit of normal motion.

Actually, all coefficients in this circuit are already known

And can be substituted into integral or equivalent differential equation

Taking Laplace transform of differential equation we can get so called transfer function

Now, using Bode magnitude plot we can get frequency response of the accelerometer as

Obtained frequency response of the accelerometer undergoing a normal motion including the effect of gas film is presented in Figure 5. As it was mentioned before, useful bandwidth has order of natural frequency of normal motion.

In the same way analysis of tilt motion can be done. Equivalent circuit is presented in Figure 6.

Figure 6. equivalent circuit of tilt motion.

It is applied everywhere that rotations around X and axes are equivalent due to symmetry.

Since governing equation is the same as for normal motion, transfer function is following

In Figure 7 obtained frequency response on tilt motion of the accelerometer is plotted.

From two obtained frequency responses for different motions of the accelerometer we can conclude that its useful bandwidth is limited by natural frequencies. Therefore, the assumption of damping behavior of gas film is always valid for designed accelerometer. Because accelerometer is actually able to measure only normal acceleration maximum allowable operation frequency of device may be set around

(according to natural frequency and frequency response).

Stability

Because both of transfer function are of the same form, both of them have no zeros and have two poles.

For normal motion poles are: