Introduction
Static model analysis
Proof mass
Suspension beams
Static deflection
Residual stress and Poisson’s ratio
Spring constants
Strain under acceleration 100 g and -100g
Sensitivity
Thermal noise
Resolution due to the ADC
Maximum acceleration
Dynamic model analysis
Etching time
Coefficients of basic equations
Natural frequencies
Damping ratios
Cut-off frequencies and squeeze numbers
Sensor system simulation
Equivalent circuits
Stability
Discussion
Introduction
In this work possible design of accelerometer, which can be produced using MOSIS 2 poly and 2 metal process, will be considered. The not in scale sketch of accelerometer is presented in Fig. 1. To etch silicon under the proof mass post-process isotropic etching will be applied and some additional mass of Al will added by wire bonding in order to make total mass 10 times of the initial mass.
To make seismic mass of sensor as big as possible we should use all available layers. All such layers are listed in Table 1.
| Layer | Thickness, µm | Density, x103 kg/m3 |
| Overglass | 1 | 2.5 |
| Metal2 | 1.15 | 2.7 |
| Ox2 | 0.65 | 2.5 |
| Metal1 | 0.6 | 2.7 |
| Ox1 | 0.85 | 2.5 |
| Poly2 | 0.4 | 2.3 |
| Polyox | 0.08 | 2.5 |
| Poly1 | 0.4 | 2.3 |
| FOX+ThinOx | 0.6 | 2.5 |
| ∑ 5.63 |
Because there are sixteen etching holes in proof mass its total area becomes:
it is taken into account here that total mass is multiplied by 10 by adding aluminum layer above.
Suspension beamsBeams are very important part of accelerometer. Because geometry is already selected we only can choose now which layers we want to use. It is clear that it’s better to use one kind of material for beams in order to avoid residual stress due to different thermal expansion coefficient. So, only silicon oxide can be used. Some of possible combinations are listed in Table 2.
| № | FOX+ThinOx | Ox1 | Ox2 | Overglass | Total thickness, µm | z, position of poly |
| 1 | Ч | Ч | 1.25 | -0.025 | ||
| 2 | Ч | Ч | 1.45 | -0.125 | ||
| 3 | Ч | Ч | 1.6 | -0.2 | ||
| 4 | Ч | Ч | Ч | Ч | 3.1 | -0.95 |
Field and thin oxide have to be used because it is only protection for polysilicon piezoresistor from bottom side. From first three rows in Table 2 we can see that parameter z increases with increasing of thickness of silicon oxide above polysilicon, because it causes bigger strain. Making absolute value of z bigger sensitivity will also increased. So the biggest sensitivity can be obtained using the thickest beam, i.e. all layers will be used. It will be shown below that with such choice of beam structure piezoresistor’s polysilicon strain under acceleration 100g is lower then critical strain for polysilicon. It means chosen design satisfies original spec for our sensor to be able to measure acceleration in range ±100g.
Static deflectionTo find static deflection of beam at x = Lb (for beams without residual stress)
we need to know spring constant Kz. For chosen geometry of sensor it can be found as follows
Deflection will be found for conditions when accelerometer is under acceleration 
and 
.
To apply further analysis we must be sure assumption of small deflection is valid.
Obtained ratio is one order less then unity, so we can consider small deflection assumption is applicable.
Residual stress and Poisson’s ratioThe residual stress in any structure is usually due to “non-ideal” fabrication. It can cause some lateral forces acting on beams. Residual stress most commonly exists when two different materials are connected together because of different thermal expansion coefficients. So, in this work, because one type of material is used for beams influence of residual stress will be neglected (as it is done in previous section for deflection). But, in general, presence of residual stress will increase or decrease effective spring constant depending on direction of acceleration.
Generally, normal stress 
and 
in beams are related to the strain 
and 
like:
where v is Poisson ratio. From equations above it can be seen that total strain can be affected by stress in normal direction. Influence of Poisson ratio may be considered in effective Young’s modulus
The correction term 
can be found from Figure 2. Taking into account that 
and 
, the aspect ratio for beam is 
and corresponding correction is actually very small. Together with small value of Poisson ratio v correction of effective Young’s modulus may not be considered. In further analysis Young’s modulus will be used without correction.
Spring constants
Spring constant for normal motion of proof mass was found earlier and equal to
Due to symmetric design of accelerometer lateral spring constants are equal and can be found from equation
Because in such configuration of sensor momentum of rotation of proof mass is zero, when we consider only normal motion, the strain can be found from equation
Figure 3. The shape of deflected beam.
From Fig. 3 it is clear that shape of deflected beam is symmetrical with respect to its central point. And the only difference is direction of curvature at edges of beam, and, subsequently, z position of polysilicon piazoresistor has different sign at different edges. So, the strains at 
and 
will just have different sign.
Where beam deflection 
under acceleration 100g was found before. For opposite acceleration strains have opposite sign respectively.
Because absolute value of strains for 100g and -100g are the same, further analysis will only due to acceleration 100g.
Sensitivity Being under acceleration piezoresistors at different edges of beam will have opposite strains and will cause opposite addition to their own resistance. Taking also into account circuit of Wheatstone bridge we can calculate voltage difference 
:
And relative changing of resistance can be obtained with the help of defined strain:
Applying that for polysilicon Gage factor is 
This is actually sensitivity under 100g acceleration. To obtain the sensitivity per unit acceleration we should do following:
And for private case of input voltage 
and acceleration 
output is expected to be
Electric noise currents in circuit are caused by electrons thermal motion in wires. These currents will affect the minimum detectable acceleration (if we consider all other are ideal). And resolution of accelerometer due to thermal noise can be found as follows:
Where 
is Boltzman constant, and 
is selected resistance of polysilicon piezoresistors, specific sensitivity 
again is for operation mode .
It was applied that sensor is operated at normal condition and 
.
As it was found in previous section, thermal noise is very small. So, another issue which should be considered in order to find resolution of our accelerometer is resolution due to used ADC. It is supposed that 16 but ADC will be used with designed sensor and it digitizes voltage in range -1.25V ~1.25V.
This error is much bigger and it will be dominant for accelerometer resolution.
Maximum accelerationPolysilicon, 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
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 frequenciesFor normal motion natural frequency is
Natural frequencies of rotation around X and Y axes are again the same because of symmetry of proof mass:
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.
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 circuitsEquivalent 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).
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
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