Design of micro accelerometer (стр. 1 из 3)
Table of content
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.
| 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 beams
Beams 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 deflection
To 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 ratio
The 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
Strain under acceleration 100 g and -100g
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 
Thermal noise
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
.Resolution due to the ADC
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.