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:

and for tilt motion:

Discussion

 

Specifications of accelerometer made using MOSIS process were estimated. Some of features are presented in Table 3. Also, corresponding specifications of ADXL50 are presented for comparison.

As we can see some of characteristics, as device size, dynamic range and bandwidth, have similar range.

These two accelerometers use different readout principles. The ADXL50 uses a capacitive measurement method. Whereas accelerometer designed in this work uses piezoresistors to generate output signal. But still comparable characteristic can be obtained. Moreover, some of parameters of accelerometer made by MOSIS process are better. For example, it has higher sensitivity and lower noise.

In this work to find some parameters sometimes very rough estimations were applied. In order to find their values more precisely more accurate techniques are required. But made analysis is suitable to see performance of a device which can be achieved if we use MOSIS process to fabricate this device.

Specification	Value	ADXL50	Unit
Device size, approx.	1x1x-	9.4x9.4x24.2	mm
Seismic mass	3.6	-
Dynamic range	-100~100	-50~50	g
	-980~980	-490~490
Sensitivity
Resolution		-
	0.66	g
Noise
			g
Frequency range	Up to 15	Up to 10	kHz

Table 3. Accelerometer’s specifications and comparison with ADXL50