In general, the capacitive sensor area can be expressed as follows:
A= 2N π r x (1)
With:
N: the CNTs number on the electrode surface,
r: the radius of one carbon nano-tube,
x: electrode displacement,
Currently; for 1 cm2, the surface density of the CNTs is equal to 109 tubes and their diameter is approximately equal to250 nm, as well as the electrodes area takes the value of 0.78 cm2 [2]. Then formula (1) becomes:
A= 612.6x (2)
The global capacitance of senoris given by the following expression [10]:
C= ϵ0 ϵrA/2d (3)
With:
ϵ0 : free space permittivity, ϵr : electrolyte relative permittivity and d : electrode-electrolyte separation.
In table 1, the dimensions and characteristics of the structural design are shown:
Table 1
The detailed structure design
Parameters
|
Value
|
Chip width
|
5 cm
|
Chip length
|
7 cm
|
d
|
1 nm
|
ϵ0
|
8.8549 *10-12 F/m
|
ϵ
|
1
|
Spring constant (K)
|
531N/m
|
The maximum capacitance
|
135.5 µF
|
The movable electrode surface
|
200mm2
|
After replacing equation 2 in equation 3 and taking the values of dimensions and characteristics from Table 1, we obtain the following formula:
C= 2.71x (4)
The modeling of the mechanical part of the capacitive sensor is made by applying Newton's second law of motion, which is given as follows [10]:
∑F = mγ = m d2x/dt2 (5)
The modelization detail is given by the work [10], and the developed model of displacement according to the mechanical parameters of the capacitive accelerometer is as follows:
x = ω2 y / ωn2 [(1 – (ω/ ωn)2)2 + (2ζω/ωn)2]1/2 (6)
Where:
x: displacement of vibratory movement
ω: relative frequency
ωn: naturel frequency
ζ: damping rate
Thefollowing equation is obtained by replacing equation(6) in equation (4):
C= 2.71 ω2 y / ωn2 [(1 – (ω/ ωn)2)2 + (2ζω/ωn)2]1/2 (7)
The acceleration measured by the capacitive sensor is expressed as follows:
a=ω2 y (8)
a=ωn2 C [(1 – (ω/ωn)2)2 + (2ζω/ωn)2]1/2 /2.71 (9)
To ensure better accuracy of the capacitive sensor, it must minimize the measurement error. To achieve this objective, the measurement error is expressed as a function of the frequency margin by the following relation and the appropriate choice of the latter minimizes this error.
E=[(d2x/dt2)/(d2y/dt2)]-1=[1/(1–(ω/ωn)2)2+(2ζω/ωn)2]1/2– 1 (10)
The simulation of this equation makes it possible to minimize the measurement error by the appropriate choice of the frequency margin and thus to improve the measurement precision.