Polymer Conductive Membrane-Based Non-Touch Mode Circular Capacitive Pressure Sensors: An Analytical Solution-Based Method for Design and Numerical Calibration

In this paper, an analytical solution-based method for the design and numerical calibration of polymer conductive membrane-based non-touch mode circular capacitive pressure sensors is presented. The accurate analytical relationship between the capacitance and applied pressure of the sensors is derived by using the analytical solution for the elastic behavior of the circular polymer conductive membranes under pressure. Based on numerical calculations using the accurate analytical relationship and the analytical solution, the analytical relationship between the pressure as output and the capacitance as input, which is necessary to achieve the capacitive pressure sensor mechanism of detecting pressure by measuring capacitance, is accurately established by least-squares data fitting. An example of how to arrive at the design and numerical calibration of a non-touch mode circular capacitive pressure sensor is first given. Then, the influence of changing design parameters such as membrane thickness and Young’s modulus of elasticity on input–output relationships is investigated, thus clarifying the direction of approaching the desired input–output relationships by changing design parameters.

The basic structure and modes of operation of a membrane elastic deflection-based capacitive pressure sensor are shown in Figure 1, where the fixed electrode plate on a substrate forms a parallel plate capacitor together with the initially flat undeflected conductive membrane (as a movable electrode plate of the capacitor). On application of pressure q, the conductive membrane elastically deflects towards the fixed electrode plate, making the initial parallel plate capacitor become a non-parallel plate capacitor and making the initial parallel plate capacitor become a non-parallel plate capacitor and resulting in a change in capacitance of the capacitor. Before the conductive membrane touches the insulator layer coating on the fixed electrode plate, the capacitive pressure sensor is said to operate in non-touch mode or normal mode and called a non-touch mode or normal mode capacitive pressure sensor [24][25][26][27][28][29], as shown in Figure 1b. Additionally, after the conductive membrane touches the insulator layer, the capacitive pressure sensor is said to operate in touch mode and called a touch mode capacitive pressure sensor [23,[30][31][32][33], as shown in Figure 1c. Obviously, the applied pressure q can be expected to be determined by measuring the capacitance of the non-parallel plate capacitor, due to their one-to-one correspondence (analytical relationship), which is the basic principle of such capacitive pressure sensors. However, the analytical relationship between the capacitance of the non-parallel plate capacitor and the applied pressure is very difficult to be exactly established due to the strong nonlinearity of the elastic behavior of the deflected conductive membrane under pressure. So, various approximation methods have to be used to obtain approximate analytical relationships between capacitance and pressure. In particular, the non-parallel plate capacitor with touch mode of operation is often simplified as an equivalent parallel plate capacitor, where only the capacitance in the touched area of the insulator layer and conductive membrane is considered and the capacitance in the untouched area is ignored [23,30,31], because the effective gap between the fixed electrode plate and conductive membrane is the thickness of the insulator layer, and the insulator layer can be designed to be very thin and have a very large dielectric constant. Furthermore, the touched area was also assumed to be approximately proportional to the applied pressure [30]. This makes it possible to establish a nearly linear analytical relationship between capacitance and pressure. On the other hand, because the non-parallel plate capacitor with non-touch mode of operation has an intrinsic nonlinear capacitance-pressure relationship, many efforts have been made to reduce its nonlinear characteristic either by modifying the shape of the fixed electrode plate [25][26][27]34] or by using special nonlinear converter circuits [29,35]. However, the existing studies often suggest that non-touch mode capacitive pressure sensors are far inferior to touch mode capacitive pressure sensors in terms of the easy However, the analytical relationship between the capacitance of the non-parallel plate capacitor and the applied pressure is very difficult to be exactly established due to the strong nonlinearity of the elastic behavior of the deflected conductive membrane under pressure. So, various approximation methods have to be used to obtain approximate analytical relationships between capacitance and pressure. In particular, the non-parallel plate capacitor with touch mode of operation is often simplified as an equivalent parallel plate capacitor, where only the capacitance in the touched area of the insulator layer and conductive membrane is considered and the capacitance in the untouched area is ignored [23,30,31], because the effective gap between the fixed electrode plate and conductive membrane is the thickness of the insulator layer, and the insulator layer can be designed to be very thin and have a very large dielectric constant. Furthermore, the touched area was also assumed to be approximately proportional to the applied pressure [30]. This makes it possible to establish a nearly linear analytical relationship between capacitance and pressure. On the other hand, because the non-parallel plate capacitor with non-touch mode of operation has an intrinsic nonlinear capacitance-pressure relationship, many efforts have been made to reduce its nonlinear characteristic either by modifying the shape of the fixed electrode plate [25][26][27]34] or by using special nonlinear converter circuits [29,35]. However, the existing studies often suggest that non-touch mode capacitive pressure sensors are far inferior to touch mode capacitive pressure sensors in terms of the easy realization of nearly linear capacitance-pressure relationships [30]. However, it should also be pointed out that the nearly linear capacitance-pressure relationships of the touch mode or non-touch Polymers 2022, 14, 3087 3 of 38 mode capacitive pressure sensors in the literature all apply only to a certain pressure range; that is, these sensors are designed to linearly operate within a certain pressure range, and beyond this pressure range, they are still nonlinear. In other words, their capacitancepressure relationships are nearly linear in a certain pressure range and, from a point of view beyond this pressure range, are still nonlinear. However, such a segment of nearly linear capacitance-pressure relationships is, in fact, not very difficult to achieve, as long as the analytical solution for the elastic behavior of the circular conductive membrane under pressure can be obtained, which can be seen later in Section 3.
In this study, an analytical solution-based method for design and numerical calibration of polymer conductive membrane-based non-touch mode circular capacitive pressure sensors is presented. The circular polymer conductive membranes are used as the pressure sensing elements, the movable electrode plates, of capacitive pressure sensors. They are usually fixed at their circular peripheries, thus will exhibit axisymmetric deformation with large deflection when subjected to a uniform differential pressure between their upper and lower opposite surfaces. By controlling the range of pressure applied, they do not touch the fixed electrode plate of the sensors so as to keep the non-touch mode of operation. Due to the fact that their upper and lower opposite surfaces are simultaneously stretched during deflection, there is no compressive stress at all but only tensile stress on their cross sections. Therefore, the elastic behavior of free deflection of the circular polymer conductive membranes under pressure can be regarded as a problem of axisymmetric deformation with large deflection of an initially flat, peripherally fixed circular membrane under uniformly distributed transverse loads. Essential to the design and numerical calibration of such non-touch mode circular capacitive pressure sensors is the analytical solutions of stress and deflection for this axisymmetric deformation problem. In this paper, they are accurately derived, and the obtained analytical solution of stress is used to determine the maximum pressure allowed to be applied to the non-touch mode circular capacitive pressure sensors, which depends on the yield strength of the circular membranes. The accurate analytical relationship between the total capacitance and applied pressure of the sensors is derived by using the analytical solution of deflection and is given in the form of the integral of the membrane deflection that is a strongly nonlinear function of the applied pressure. Therefore, in order to achieve the capacitive pressure sensor mechanism of detecting pressure by measuring capacitance, the accurate analytical relationship between the pressure as output and the capacitance as input is given by using the least-squares data fitting based on numerical calculations.
The analytical solution-based method presented here can make the non-touch mode circular capacitive pressure sensors be more accurately designed and numerically calibrated, thus greatly reducing the dependence on experimental calibrations. In comparison with the methods in the literature such as modifying the shape of substrate electrode plates [25][26][27]34] or using special nonlinear converter circuits [29,35], this novel method has the advantages of intuition, clarity, strong tunability and operability. By changing design parameters, including geometric parameters (such as the thickness of the circular membranes and the initial gap between initially flat undeflected circular membranes and fixed electrode plates) and physical parameters (such as the Poisson's ratio and Young's modulus of elasticity of the circular membranes), it can easily realize the accurate analytical relationships between the pressure as output and the capacitance as input, including linear and non-linear relationships. Therefore, from this point of view, the view in the literature is open to debate that non-touch mode capacitive pressure sensors are far inferior to touch mode capacitive pressure sensors in the easy realization of nearly linear input-output relationships [30]. This should be due to the lack of the exact analytical solutions and their effective applications.
The paper is organized as follows. In the following section, the accurate analytical relationship between the total capacitance and applied pressure of the non-touch mode circular capacitive pressure sensors is derived in detail, the analytical solutions of stress and deflection for the elastic behavior of free deflection of the circular conductive membranes under pressure are accurately derived, and how to design and numerically calibrate the non-touch mode circular capacitive pressure sensors is described in detail. In Section 3, an example is first given of how to arrive at a design and numerical calibration of nontouch mode circular capacitive pressure sensors. Then, in order to clarify the direction of approaching the desired pressure-capacitance relationships by changing design parameters, the influence of changing design parameters on pressure-capacitance relationships is investigated. Concluding remarks are given in Section 4.

Materials and Methods
The geometry and configuration of a non-touch mode circular capacitive pressure sensor is shown in Figure 2a, where the initially flat, undeflected, circular conductive membrane with Poisson's ratio v, Young's modulus of elasticity E, thickness h and radius a forms a parallel plate capacitor together with the electrode plate fixed to the substrate, t denotes the thickness of the insulator layer coating on the substrate electrode plate, and g denotes the initial gap between the insulator layer and the initially flat, undeflected, circular conductive membrane. On application of pressure (the uniformly distributed transverse loads q), as shown in Figure 2b, the initially flat, undeflected, circular conductive membrane deflects towards the substrate electrode plate, making the initial parallel plate capacitor become a non-parallel plate capacitor and resulting in a change in capacitance of the capacitor. In Figure 2b, the dash-dotted line represents the plane in which the geometric middle plane of the initially flat, undeflected, circular conductive membrane is located, o denotes the origin of the introduced cylindrical coordinate system (r, ϕ, w), r is the radial coordinate, ϕ is the angle coordinate but not represented in Figure 2b, and w is the axial coordinate and denotes the deflection of the deflected conductive membrane.
The paper is organized as follows. In the following section, the accurate analytical relationship between the total capacitance and applied pressure of the non-touch mode circular capacitive pressure sensors is derived in detail, the analytical solutions of stress and deflection for the elastic behavior of free deflection of the circular conductive membranes under pressure are accurately derived, and how to design and numerically calibrate the non-touch mode circular capacitive pressure sensors is described in detail. In Section 3, an example is first given of how to arrive at a design and numerical calibration of non-touch mode circular capacitive pressure sensors. Then, in order to clarify the direction of approaching the desired pressure-capacitance relationships by changing design parameters, the influence of changing design parameters on pressure-capacitance relationships is investigated. Concluding remarks are given in Section 4.

Materials and Methods
The geometry and configuration of a non-touch mode circular capacitive pressure sensor is shown in Figure 2a, where the initially flat, undeflected, circular conductive membrane with Poisson's ratio v, Young's modulus of elasticity E, thickness h and radius a forms a parallel plate capacitor together with the electrode plate fixed to the substrate, t denotes the thickness of the insulator layer coating on the substrate electrode plate, and g denotes the initial gap between the insulator layer and the initially flat, undeflected, circular conductive membrane. On application of pressure (the uniformly distributed transverse loads q), as shown in Figure 2b, the initially flat, undeflected, circular conductive membrane deflects towards the substrate electrode plate, making the initial parallel plate capacitor become a non-parallel plate capacitor and resulting in a change in capacitance of the capacitor. In Figure 2b, the dash-dotted line represents the plane in which the geometric middle plane of the initially flat, undeflected, circular conductive membrane is located, o denotes the origin of the introduced cylindrical coordinate system (r, φ, w), r is the radial coordinate, φ is the angle coordinate but not represented in Figure 2b, and w is the axial coordinate and denotes the deflection of the deflected conductive membrane.  Before the pressure q is applied to the circular conductive membrane, the total initial capacitance C 0 of the initial parallel plate capacitor formed by the initially flat, undeflected, circular conductive membrane and the substrate electrode plate comprises the capacitance C 1 and C 2 of two series parallel plate capacitors, where C 1 refers to the capacitance of the parallel plate capacitor with the insulator layer gap t and relative permittivity ε r1 , and C 2 refers to the capacitance of the parallel plate capacitor with the air gap g and relative permittivity ε r2 . Therefore, if the vacuum permittivity is denoted by ε 0 , then where and Thus, After the pressure q is applied to the conductive membrane, the total capacitance C of the non-parallel plate capacitor formed by the deflected circular conductive membrane and the substrate electrode plate is still composed of the capacitance of two series capacitors: one is the capacitance C 1 of the parallel plate capacitor with the insulator layer gap t and relative permittivity ε r1 , which is still given by Equation (2); the other is the capacitance C 2 of the air dielectric non-parallel plate capacitor with the relative permittivity ε r2 and uneven distribution of air gap g-w(r) (see Figure 2b). Therefore, the expression of capacitance C 2 needs to be further derived. To this end, let us take a micro area element, ABCD, from the substrate electrode plate, as shown in Figure 3.
pacitance C1 and C2 of two series parallel plate capacitors, where C1 refers to the capacitance of the parallel plate capacitor with the insulator layer gap t and relative permittivity εr1, and C2 refers to the capacitance of the parallel plate capacitor with the air gap g and relative permittivity εr2. Therefore, if the vacuum permittivity is denoted by ε0, then where ε ε π and ε ε π Thus, ε ε π ε ε π ε ε ε π ε ε ε ε π ε ε π After the pressure q is applied to the conductive membrane, the total capacitance C of the non-parallel plate capacitor formed by the deflected circular conductive membrane and the substrate electrode plate is still composed of the capacitance of two series capacitors: one is the capacitance C1 of the parallel plate capacitor with the insulator layer gap t and relative permittivity εr1, which is still given by Equation (2); the other is the capacitance ′ 2 C of the air dielectric non-parallel plate capacitor with the relative permittivity εr2 and uneven distribution of air gap g-w(r) (see Figure 2b). Therefore, the expression of capacitance ′ 2 C needs to be further derived. To this end, let us take a micro area element, ABCD, from the substrate electrode plate, as shown in Figure 3.  The area of the micro area element ABCD is After ignoring the higher-order terms (the second term in Equation (5)), ∆S can be approximated by r∆r∆ϕ, while the air gap between this micro area element ABCD on the substrate electrode plate and the corresponding deflected conductive membrane can be approximated by g − w(r), resulting in and dr.
Thus, the total capacitance C of the non-parallel plate capacitor formed by the deflected circular conductive membrane and the substrate electrode plate may finally be written as It can be seen from Equation (8) that the total capacitance C can be determined as long as an analytical expression for deflection w(r) is available. Therefore, the analytical solutions of deflection w(r) and stress σ r (r) of the deflected circular conductive membrane under pressure q is vital to the determination of the total capacitance C of the non-parallel plate capacitor formed by the deflected circular conductive membrane under pressure q and the substrate electrode plate.
To this end, we have to analytically solve the problem of axisymmetric deformation with large deflection of the deflected circular conductive membrane under the uniformly distributed transverse loads q. However, for the sake of coherence, the detailed derivation of the analytical solution of this axisymmetric deformation problem is arranged in the Appendix A. The analytical expressions for stress σ r (r) and deflection w(r) can be written as, from Equations (A16), (A22) and (A23), and where c 2i and b 2i are the coefficients of the power series, which are listed in Appendix B. It can be seen from Appendix B that when i = 0 the coefficients c 2i and b 2i are expressed into the polynomials with regard to the coefficients b 0 , Poisson's ratio v and dimensionless parameter Q (the dimensionless pressure, see Equation (A16)). The coefficients b 0 and c 0 are usually called undetermined constants. For a given Poisson's ratio v, Young's modulus of elasticity E, thickness h, radius a and pressure q, the undetermined constant b 0 can be determined by solving Equation (A24). Additionally, with the known b 0 , all the coefficients c 2i and b 2i when i = 0 can be determined (see Appendix B), such that the undetermined constant c 0 can be determined by Equation (A25). In this way, the deflection expression, i.e., Equation (10), can be determined due to the known coefficient c 2i (i = 0, 1, 2, 3 . . . ). The maximum stress σ m and maximum deflection w m of the axisymmetrically deflected circular conductive membrane are at its center (i.e., at r = 0), hence given by and w m = ac 0 .
For a given conductive membrane (given Poisson's ratio v, Young's modulus of elasticity E, thickness h, radius a and yield strength σ y ), the maximum stress σ m at any pressure q can be determined by Equation (11). To ensure the strength of the material, it is assumed that the working stress of the conductive membrane is always controlled below 70% of the yield strength σ y . So, if the pressure q at σ m = 0.7σ y is equal to the maximum pressure of a given pressure measurement range, then the given conductive membrane meets the design requirements; otherwise, a new conductive membrane (with different design parameters such as membrane thickness h, Poisson's ratio v and Young's modulus of elasticity E) needs to be selected. On the other hand, the maximum deflection w m at σ m = 0.7σ y can be determined by Equation (12) and is used primarily to determine the initial gap g between the insulator layer and the initially flat, undeflected, circular conductive membrane, see Figure 2a. The minimum value of the initial gap g should be greater than but as close as possible to this maximum deflection w m .
After plugging the known deflection expression (i.e., for given Poisson's ratio v, Young's modulus of elasticity E, thickness h, radius a and pressure q, the power series coefficients c 2i /a 2i−1 in Equation (10) are known) into Equation (8), the total capacitance C of the non-parallel plate capacitor, which is formed by the deflected circular conductive membrane under the given pressure q and the substrate electrode plate, can finally be determined with the known initial gap g, vacuum permittivity ε 0 , and relative permittivities ε r1 and ε r2 . In this way, a pair of numerical values of calculated capacitance C and given loads q, having an intrinsic analytical relationship, is thus established. Additionally, with another given value of pressure q, another pair of numerical values of calculated capacitance C and given loads q can be further established.
Therefore, the numerical calculations of a progressive increase in the values of pressure q will result in a data sequence (sequential number pairs) with respect to numerical values of calculated capacitance C and given loads q, as shown in the next section. Additionally, further, based on this data sequence, the analytical relationship between loads q and capacitance C can be established by using least-squares data fitting, including straight line fitting and curve fitting, as shown in the next section. In each fitting function, the ranges of variation of loads q and capacitance C are affected by different requirements of fitting accuracy (average sum of fitting error squares). On the other hand, for given requirements of fitting accuracy, the ranges of variation of loads q and capacitance C can also be changed by changing geometric parameters (such as the thickness h and radius a of the conductive membranes and the initial gap g) and physical parameters (such as the Poisson's ratio v and Young's modulus of elasticity E of the conductive membranes), as shown in Section 3.2.
All in all, with Equation (8) and the analytical solution in Appendix A, the nontouch mode circular capacitive pressure sensors can be perfectly designed and numerically calibrated, thus greatly reducing the dependence on experimental calibration.

Results and Discussion
In this section, an example is first given of how to use Equation (8) and the analytical solution in Appendix A to realize the design and numerical calibration of non-touch mode circular capacitive pressure sensors (see Section 3.1). Then, in order to clarify the direction of approaching the desired pressure-capacitance relationships by changing design parameters, the influence of changing design parameters on pressure-capacitance relationships is comprehensively investigated, such as changing the initial gap g between the insulator layer coating on the substrate electrode plate and the initially flat undeflected circular conductive membrane, the thickness h of the circular conductive membranes, Young's modulus of elasticity E, Poisson's ratio v and the thickness t of the insulator layers, see Section 3.2.
In fact, Equation (8) has given the accurate analytical relationship between the capacitance C and the pressure q, where q is included in the power series coefficients c 2i of the deflection w(r) (see Appendix B). However, in order to achieve the sensor mechanism of detecting pressure by measuring capacitance, we need to know the accurate analytical relationship between the pressure q as output and the capacitance C as input, that is, the analytical expression of the capacitance C as independent variable and the pressure q as dependent variable, q(C). Obviously, such an analytical expression cannot be directly given due to the strong nonlinearity between the deflection w(r) and the applied pressure q. Therefore, in this case, we have to perform a lot of numerical calculations using Equation (8) and the analytical solution of deflection and use least-squares data fitting to arrive at the analytical expression q(C), which may be seen in Section 3.1.
On the other hand, the numerical calculations using Equation (8) and the analytical solution of deflection can only be carried out on the premise that the circular conductive membrane is known and the range of pressure q is specified. Therefore, the design of a non-touch mode circular capacitive pressure sensor whose pressure range is beforehand specified has to begin with a tentative choice of a circular conductive membrane, including membrane thickness h, Poisson's ratio v and Young's modulus of elasticity E. If the resulting pressure-capacitance relationship, q(C), does not satisfy the desired usage or technical requirements, especially the range of the input capacitance C and output pressure q, then the design parameters, especially the membrane thickness h and Young's modulus of elasticity E, must be adjusted. Section 3.2 gives the direction of the adjustment for approaching the desired usage or technical requirements.

An Example of Design and Numerical Calibration Based on Analytical Solutions
A non-touch mode circular capacitive pressure sensor is assumed to use a circular conductive membrane with Poisson's ratio v = 0.47, Young's modulus of elasticity E = 7.84 MPa, radius a = 100 mm, thickness h = 1 mm and yield strength σ y = 2.4 MPa. The maximum value of the applied pressure q can be determined by the condition that the maximum stress σ m of the circular conductive membrane under pressure q does not exceed its yield strength σ y = 2.4 MPa. Table 1 shows the calculation results as the applied pressure q progressively increases, where the undetermined constants b 0 and c 0 are calculated by Equations (A24) and (A25), the maximum stress σ m and maximum deflection w m are calculated by Equations (11) and (12). It may be seen from Table 1      If the working stress of the circular conductive membrane is always controlled to be less than or equal to 70% of the yield strength σy, that is, σm ≤ 0.7 σy ≈ 1.68 MPa, then it can be seen from Table 1 that the maximum operation pressure should not exceed 21.225 KPa. Therefore, the values of the undetermined constants b0 at pressures less than or equal to     If the working stress of the circular conductive membrane is always controlled to be less than or equal to 70% of the yield strength σy, that is, σm ≤ 0.7 σy ≈ 1.68 MPa, then it can be seen from Table 1 that the maximum operation pressure should not exceed 21.225 KPa. Therefore, the values of the undetermined constants b0 at pressures less than or equal to If the working stress of the circular conductive membrane is always controlled to be less than or equal to 70% of the yield strength σ y , that is, σ m ≤ 0.7 σ y ≈ 1.68 MPa, then it can be seen from Table 1 that the maximum operation pressure should not exceed 21.225 KPa. Therefore, the values of the undetermined constants b 0 at pressures less than or equal to 21.225 KPa in Table 1 will be used to determine the values of the coefficients c 2i (see Appendix B for their expressions), as shown in Tables 2 and 3. Moreover, from Table 1, we may also see that the value of the maximum deflection w m corresponding to 21.225 KPa pressure is about 39.67 mm. Therefore, the initial gap g between the initially flat undeflected conductive membrane and the insulator layer coating on the substrate electrode plate should be greater than or equal to 41 mm. For investigating the influence of changing the initial gap g on the input-output relationship between the input capacitance C and the output pressure q, the pressure-capacitance relationship q(C), here, the initial gap g takes 41 mm, 46 mm and 51 mm, respectively.
If the insulator layer is assumed to take 0.1 mm of polystyrene, then t = 0.1 mm and the relative permittivity ε r1 = 2.5. In addition, the vacuum permittivity ε 0 = 8.854 × 10 −12 F/m = 8.854 × 10 −3 pF/mm, and the air relative permittivity ε r2 = 1.00053. The deflection expressions describing the shape of the deflected conductive membrane under different pressures q can be determined by Equation (10) with the values of the coefficients c 2i in Tables 2 and 3. Therefore, the values of the total capacitance (at rest) of the non-parallel plate capacitor formed by the deflected circular conductive membrane and the substrate electrode plate may finally be determined by Equation (8), which are listed in Table 4, where the definite integral in Equation (8) was calculated by using Maple 2018 software package. Figure 6 shows the variations of pressure q with capacitance C, showing that the increase in the initial gap g will increase the degree of linearity of the pressure-capacitance relationship q(C). From this point of view, the view in the literature is open to debate that non-touch mode capacitive pressure sensors are far inferior to touch mode capacitive pressure sensors in the easy realization of nearly linear input-output relationships [30]. The linearization in such a way, however, will narrow the range of the input capacitance and eventually increase the output pressure per unit capacitance, in addition to increasing the edge effect in capacitance of the non-parallel plate capacitor. So, it is best not to do so unless necessary. In fact, it can be imagined from Figure 6 that the nearly linear pressurecapacitance relationship q(C) can also be realized by least-squares data fitting of the data for g = 41 mm. Figure 7 shows the results of least-squares fitting, where Functions 1-4 are the results for g = 41 mm, Function 5 is the result for g = 46 mm, Function 6 is the result for g = 51 mm and Functions 1, 5 and 6 are fitted by straight lines, and Function 2 is fitted by a quadratic function, Function 3 by a cubic function and Function 4 by a quartic function. The resulting fitting functions are listed in Table 5, and the average sum of fitting error squares of each fitting function is shown in the footer of Table 5.    As can be seen from Table 5 and Figure 7, the above design and numerical calibration can realize five non-touch mode circular capacitive pressure sensors with different pressure-capacitance relationships, two linear (Functions 1 and 6) and three nonlinear (Functions 2-4). Obviously, Function 1 should be preferred to Function 6 if a 1~8 KPa pressure range is sufficient for use, because the output pressure per unit capacitance is about 1.940 KPa/pF for Function 1 but 4.267 KPa/pF for Function 6 (which are calculated from Table 5). However, for today's advanced digital technologies, the emphasis on nearly linear input-output relationships makes no sense, because in most cases, using digital technologies is feasible. Therefore, in this sense, Function 4 should be one of the best choices for pressure monitoring microcomputer systems based on such non-touch mode circular capacitive pressure sensing devices.   As can be seen from Table 5 and Figure 7, the above design and numerical calibration can realize five non-touch mode circular capacitive pressure sensors with different pressure-capacitance relationships, two linear (Functions 1 and 6) and three nonlinear (Functions 2-4). Obviously, Function 1 should be preferred to Function 6 if a 1~8 KPa pressure range is sufficient for use, because the output pressure per unit capacitance is about 1.940 KPa/pF for Function 1 but 4.267 KPa/pF for Function 6 (which are calculated from Table   0  2  4  6  8  10  12  14  16  18  20  22  24  26 28 0 Least-squares fitting of the relationships between q and C in Figure 6. Table 5. The range of pressure q and capacitance C, and the analytical expressions of Functions 1-6 in Figure 7. Of course, Functions 1-4 and 6 may also not satisfy the usage or technical requirements of the input capacitance and output pressure under consideration. In this case, the design parameters, other than the initial gap g, should further be adjusted to meet the desired requirements, as shown in the next section.

Parametric Analysis
As mentioned above, although the increase in the initial gap g between the initially flat undeflected conductive membrane and the substrate electrode plate can increase the degree of linearity of the analytical relationship between input capacitance C and output pressure q, it is not a preferred option to encourage adoption. On the other hand, however, we should also see that decreasing the initial gap g can increase the range of input capacitance C, see Figure 6. The main purpose of this section is to show the influence of changing the design parameters other than the initial gap g on the analytical relationship between input capacitance C and output pressure q. To this end, we take the design parameters used in Section 3.1 as reference and change each parameter one by one on this basis, such as changing the thickness h of the conductive membranes, Young's modulus of elasticity E, Poisson's ratio v, and the thickness t of insulator layers.

Effect of Membrane Thickness on Input-Output Relationships
The design parameters used in Section 3.1 are used as reference, that is, Poisson's ratio v = 0.47, Young's modulus of elasticity E = 7.84 MPa, circular conductive membrane radius a = 100 mm, circular conductive membrane thickness h = 1 mm, insulator layer thickness t = 0.1 mm, vacuum permittivity ε 0 = 8.854 × 10 −12 F/m = 8.854 × 10 −3 pF/mm, air relative permittivity ε r2 = 1.00053, insulator layer relative permittivity ε r1 = 2.5, membrane yield stress σ y = 2.4 MPa and membrane maximum stress σ m ≤ 0.7 σ y ≈ 1.68 MPa. In this section, the thickness h of the circular conductive membrane is first increased from the reference thickness of 1 mm to 1.5 mm and then is further increased to 2 mm. When h = 1.5 mm, the calculation results are listed in Table 6, the relationships between input capacitance C and output pressure q are shown in Figure 8, the results of least-squares fitting are shown in Figure 9, the fitting functions are listed in Table 7, and the average sum of fitting error squares of each fitting function is shown in the footer of Table 7. When h = 2 mm, the calculation results are listed in Table 8, the input-output relationships are shown in Figure 10, the results of least-squares fitting are shown in Figure 11, the fitting functions are listed in Table 9, and the average sum of fitting error squares of each fitting function is shown in the footer of Table 9. The effects of an increase in the membrane thickness h from 1 mm to 1.5 mm and then to 2 mm on the fitting functions (Functions 1-4) are summarized in Figures 12-15.  Table 7. The range of pressure q and capacitance C, and the analytical expressions of the Functions 1-6 in Figure 9.   . Least-squares fitting of the relationships between q and C in Figure 8.    . Least-squares fitting of the relationships between q and C in Figure 8.     . Least-squares fitting of the relationships between q and C in Figure 10. Table 9. The range of pressure q and capacitance C, and the analytical expressions of Functions 1-6 in Figure 11.  Figure 11. Least-squares fitting of the relationships between q and C in Figure 10. Table 9. The range of pressure q and capacitance C, and the analytical expressions of Functions 1-6 in Figure 11.           It can be seen from Figures 12-15 that the change in the membrane thickness h only affects the range of output pressure q (increasing with the increase in the membrane thickness h) and does not affect the range of input capacitance C on the premise of ensuring the basically same fitting accuracy (the average sum of fitting error squares of each fitting function (e.g., Function 1, 2, 3 or 4) is basically the same (see the footers of Tables 5, 7 and 9)). It should also be noted, however, that an increase in the membrane thickness h increases the range of output pressure q, but it also moderately increases the output pressure per unit capacitance because the input capacitance C remains constant. For instance, as the membrane thickness h increases from the reference value of 1 mm to 1.5 mm and then to 2 mm, the output pressure per unit capacitance of Function 1 increases from 1.940 KPa/pF to 2.840 KPa/pF and then to 3.724 KPa/pF, while the output pressure per unit capacitance of Function 4 increases from 1.071 KPa/pF to 1.607 KPa/pF and then to 2.143 KPa/pF, which are calculated from Tables 5, 7 and 9.

Effect of Young's Modulus of Elasticity on Input-Output Relationships
The design parameters used in Section 3.1 are still used as reference, that is, v = 0.47, E = 7.84 MPa, a = 100 mm, h = 1 mm, t = 0.1 mm, ε0 = 8.854 × 10 −12 F/m = 8.854 × 10 −3 pF/mm, εr1 = 2.5, εr2 = 1.00053, σy = 2.4 MPa and σm ≤ 0.7 σy ≈ 1.68 MPa. In this section, the Young's modulus of elasticity E of the conductive membrane is first decreased from the reference value of 7.84 MPa to 5 MPa and then further decreased to 2.5 MPa. When E = 5 MPa, the calculation results are listed in Table 10, the relationships between input capacitance C and output pressure q are shown in Figure 16, the results of least-squares fitting are shown in Figure 17, the fitting functions are listed in Table 11, and the average sum of fitting error squares of each fitting function is shown in the footer of Table 11. When E = 2.5 MPa, the calculation results are listed in Table 12, the input-output relationships are shown in Figure 18, the results of least-squares fitting are shown in Figure 19, the fitting functions are listed in Table 13, and the average sum of fitting error squares of each fitting function is shown in the footer of Table 13. The effects of a decrease in the Young's modulus of elasticity E from 7.84 MPa to 5 MPa and then to 2.  It can be seen from Figures 12-15 that the change in the membrane thickness h only affects the range of output pressure q (increasing with the increase in the membrane thickness h) and does not affect the range of input capacitance C on the premise of ensuring the basically same fitting accuracy (the average sum of fitting error squares of each fitting function (e.g., Function 1, 2, 3 or 4) is basically the same (see the footers of Tables 5, 7 and 9)). It should also be noted, however, that an increase in the membrane thickness h increases the range of output pressure q, but it also moderately increases the output pressure per unit capacitance because the input capacitance C remains constant. For instance, as the membrane thickness h increases from the reference value of 1 mm to 1.5 mm and then to 2 mm, the output pressure per unit capacitance of Function 1 increases from 1.940 KPa/pF to 2.840 KPa/pF and then to 3.724 KPa/pF, while the output pressure per unit capacitance of Function 4 increases from 1.071 KPa/pF to 1.607 KPa/pF and then to 2.143 KPa/pF, which are calculated from Tables 5, 7 and 9.

Effect of Young's Modulus of Elasticity on Input-Output Relationships
The design parameters used in Section 3.1 are still used as reference, that is, v = 0.47, E = 7.84 MPa, a = 100 mm, h = 1 mm, t = 0.1 mm, ε 0 = 8.854 × 10 −12 F/m = 8.854 × 10 −3 pF/mm, ε r1 = 2.5, ε r2 = 1.00053, σ y = 2.4 MPa and σ m ≤ 0.7 σ y ≈ 1.68 MPa. In this section, the Young's modulus of elasticity E of the conductive membrane is first decreased from the reference value of 7.84 MPa to 5 MPa and then further decreased to 2.5 MPa. When E = 5 MPa, the calculation results are listed in Table 10, the relationships between input capacitance C and output pressure q are shown in Figure 16, the results of least-squares fitting are shown in Figure 17, the fitting functions are listed in Table 11, and the average sum of fitting error squares of each fitting function is shown in the footer of Table 11. When E = 2.5 MPa, the calculation results are listed in Table 12, the input-output relationships are shown in Figure 18, the results of least-squares fitting are shown in Figure 19, the fitting functions are listed in Table 13, and the average sum of fitting error squares of each fitting function is shown in the footer of Table 13. The effects of a decrease in the Young's modulus of elasticity     Least-squares fitting of the relationships between q and C in Figure 16. Table 11. The range of pressure q and capacitance C, and the analytical expressions of the fitting functions in Figure 17. Figure 17. Least-squares fitting of the relationships between q and C in Figure 16. Table 11. The range of pressure q and capacitance C, and the analytical expressions of the fitting functions in Figure 17.      Least-squares fitting of the relationships between q and C in Figure 18. Table 13. The range of pressure q and capacitance C, and the analytical expressions of the fitting functions in Figure 19.  Figure 19. Least-squares fitting of the relationships between q and C in Figure 18. Table 13. The range of pressure q and capacitance C, and the analytical expressions of the fitting functions in Figure 19.           Figures 20-23, it can be seen that the change in the Young's modulus of elasticity E affects both the range of output pressure q (increasing with the decrease in the Young's modulus of elasticity E) and the range of input capacitance C (decreasing with the decrease in the Young's modulus of elasticity E) on the premise of ensuring the basically same fitting accuracy (the average sum of fitting error squares of each fitting function (e.g., Function 1, 2, 3 or 4) is basically the same (see the footers of Tables 5, 11 and 13)). Therefore, as the Young's modulus of elasticity E decreases from the reference value of 7.84 MPa to 5 MPa and then to 2.5 MPa, the output pressure per unit capacitance of Function 1 increases from 1.940 KPa/pF to 2.633 KPa/pF and then to 4.168 KPa/pF, while the output pressure per unit capacitance of Function 4 increases from 1.071 KPa/pF to 1.402 KPa/pF and then to 1.736 KPa/pF, which are calculated from Tables 5, 11 and 13.

Effect of Poisson's Ratio on Input-Output Relationships
The design parameters used in Section 3.1 are still used as reference, that is, v = 0.47, E = 7.84 MPa, a = 100 mm, h = 1 mm, t = 0.1 mm, ε 0 = 8.854 × 10 −12 F/m = 8.854 × 10 −3 pF/mm, ε r1 = 2.5, ε r2 = 1.00053, σ y = 2.4 MPa and σ m ≤ 0.7 σ y ≈ 1.68 MPa. In this section, the Poisson's ratio v of the conductive membrane is first decreased from the reference value of 0.47 (for such as polymer films) to 0.32 (for such as metal films) and then further decreased to 0.16 (for such as graphene films). When v = 0.32, the calculation results are listed in Table 14, the relationships between input capacitance C and output pressure q are shown in Figure 24, the results of least-squares fitting are shown in Figure 25, the fitting functions are listed in Table 15, and the average sum of fitting error squares of each fitting function is shown in the footer of Table 15. When v = 0.16, the calculation results are listed in Table 16, the input-output relationships are shown in Figure 26, the results of least-squares fitting are shown in Figure 27, the fitting functions are listed in Table 17, and the average sum of fitting error squares of each fitting function is shown in the footer of Table 17 Table 15. The range of pressure q and capacitance C, and the analytical expressions of the fitting functions in Figure 25.       Figure 25. Least-squares fitting of the relationships between q and C in Figure 24.    Figure 26. Table 17. The range of pressure q and capacitance C, and the analytical expressions of the fitting functions in Figure 27.   Figure 27. Least-squares fitting of the relationships between q and C in Figure 26. Table 17. The range of pressure q and capacitance C, and the analytical expressions of the fitting functions in Figure 27.  Figure 27. Least-squares fitting of the relationships between q and C in Figure 26. Table 17. The range of pressure q and capacitance C, and the analytical expressions of the fitting functions in Figure 27.    As can be seen from Figures 28-31, especially from Figure 31, the change of the Poisson's ratio v from 0.47 to 0.32 and then to 0.16 results in only a small nearly parallel shift of the q(C) curves along the horizontal coordinate axis; that is, such a large change in the Poisson's ratio v from 0.47 to 0.32 and then to 0.16 does not have much effect on both the range of output pressure q and the range of input capacitance C. This means that when choosing a polymer conductive membrane as the movable electrode plate of a capacitor in a non-touch mode circular capacitive pressure sensor, it is sufficient to know the approximate range of Poisson's ratio rather than its exact value.   As can be seen from Figures 28-31, especially from Figure 31, the change of the Poisson's ratio v from 0.47 to 0.32 and then to 0.16 results in only a small nearly parallel shift of the q(C) curves along the horizontal coordinate axis; that is, such a large change in the Poisson's ratio v from 0.47 to 0.32 and then to 0.16 does not have much effect on both the range of output pressure q and the range of input capacitance C. This means that when choosing a polymer conductive membrane as the movable electrode plate of a capacitor in a non-touch mode circular capacitive pressure sensor, it is sufficient to know the approximate range of Poisson's ratio rather than its exact value.   As can be seen from Figures 28-31, especially from Figure 31, the change of the Poisson's ratio v from 0.47 to 0.32 and then to 0.16 results in only a small nearly parallel shift of the q(C) curves along the horizontal coordinate axis; that is, such a large change in the Poisson's ratio v from 0.47 to 0.32 and then to 0.16 does not have much effect on both the range of output pressure q and the range of input capacitance C. This means that when choosing a polymer conductive membrane as the movable electrode plate of a capacitor in a non-touch mode circular capacitive pressure sensor, it is sufficient to know the approximate range of Poisson's ratio rather than its exact value. As can be seen from Figures 28-31, especially from Figure 31, the change of the Poisson's ratio v from 0.47 to 0.32 and then to 0.16 results in only a small nearly parallel shift of the q(C) curves along the horizontal coordinate axis; that is, such a large change in the Poisson's ratio v from 0.47 to 0.32 and then to 0.16 does not have much effect on both the range of output pressure q and the range of input capacitance C. This means that when choosing a polymer conductive membrane as the movable electrode plate of a capacitor in a non-touch mode circular capacitive pressure sensor, it is sufficient to know the approximate range of Poisson's ratio rather than its exact value.

Effect of Insulator Layer Thickness on Input-Output Relationships
The design parameters used in Section 3.1 are still used as reference, that is, v = 0.47, E = 7.84 MPa, a = 100 mm, h = 1 mm, t = 0.1 mm, ε 0 = 8.854 × 10 −12 F/m = 8.854 × 10 −3 pF/mm, ε r1 = 2.5, ε r2 = 1.00053, σ y = 2.4 MPa and σ m ≤ 0.7 σ y ≈ 1.68 MPa. In this section, the thickness t of the insulator layer is first increased from the reference value of 0.1 mm to 1 mm and then to 10 mm. When t = 1 mm, the calculation results are listed in Table 18, the relationships between input capacitance C and output pressure q are shown in Figure 32, the results of least-squares fitting are shown in Figure 33, the fitting functions are listed in Table 19, and the average sum of fitting error squares of each fitting function are shown in the footer of Table 19. When t = 10 mm, the calculation results are listed in Table 20, the input-output relationships are shown in Figure 34, the results of least-squares fitting are shown in Figure 35, the fitting functions are listed in Table 21, and the average sum of fitting error squares of each fitting function are shown in the footer of Table 21. The effects of an increase in the thickness t of the insulator layer from 0.1 mm to 1 mm and then to 10 mm on the fitting functions (Functions 1-4) are summarized in Figures 36-39. The design parameters used in Section 3.1 are still used as reference, that is, v = 0.47, E = 7.84 MPa, a = 100 mm, h = 1 mm, t = 0.1 mm, ε0 = 8.854 × 10 −12 F/m = 8.854 × 10 −3 pF/mm, εr1 = 2.5, εr2 = 1.00053, σy = 2.4 MPa and σm ≤ 0.7 σy ≈ 1.68 MPa. In this section, the thickness t of the insulator layer is first increased from the reference value of 0.1 mm to 1 mm and then to 10 mm. When t = 1 mm, the calculation results are listed in Table 18, the relationships between input capacitance C and output pressure q are shown in Figure 32, the results of least-squares fitting are shown in Figure 33, the fitting functions are listed in Table  19, and the average sum of fitting error squares of each fitting function are shown in the footer of Table 19. When t = 10 mm, the calculation results are listed in Table 20, the inputoutput relationships are shown in Figure 34, the results of least-squares fitting are shown in Figure 35, the fitting functions are listed in Table 21, and the average sum of fitting error squares of each fitting function are shown in the footer of Table 21. The effects of an increase in the thickness t of the insulator layer from 0.1 mm to 1 mm and then to 10 mm on the fitting functions (Functions 1-4) are summarized in Figures 36-39.    Figure 32. Table 19. The range of pressure q and capacitance C, and the analytical expressions of the fitting functions in Figure 33.   Figure 33. Least-squares fitting of the relationships between q and C in Figure 32. Table 19. The range of pressure q and capacitance C, and the analytical expressions of the fitting functions in Figure 33.   Table 21. The range of pressure q and capacitance C, and the analytical expressions of the fitting functions in Figure 35.   Least-squares fitting of the relationships between q and C in Figure 34. Table 21. The range of pressure q and capacitance C, and the analytical expressions of the fitting functions in Figure 35.    Least-squares fitting of the relationships between q and C in Figure 34. Table 21. The range of pressure q and capacitance C, and the analytical expressions of the fitting functions in Figure 35.        From Figures 36-39, it can be seen that increasing the thickness t of the insulator layer has no effect on the range of output pressure q, and it only reduces the range of input capacitance C, resulting in an increase in the output pressure per unit capacitance. Taking Function 4 as an example, when the thickness t of the insulator layer increases from 0.1 mm to 10 mm, the output pressure per unit capacitance increases from 1.071 KPa/pF (calculated from Table 5) to 1.620 KPa/pF (calculated from Table 21). As a result, it is generally welcome for the thickness t of the insulator layer to be as thin as possible. From Figures 36-39, it can be seen that increasing the thickness t of the insulator layer has no effect on the range of output pressure q, and it only reduces the range of input capacitance C, resulting in an increase in the output pressure per unit capacitance. Taking Function 4 as an example, when the thickness t of the insulator layer increases from 0.1 mm to 10 mm, the output pressure per unit capacitance increases from 1.071 KPa/pF (calculated from Table 5) to 1.620 KPa/pF (calculated from Table 21). As a result, it is generally welcome for the thickness t of the insulator layer to be as thin as possible.

Concluding Remarks
In this paper, an analytical solution-based method for the design and numerical calibration of polymer conductive membrane-based non-touch mode circular capacitive pressure sensors is presented. This novel method can provide effective theoretical support for the design and fabrication of such sensors. From this study, the following conclusions can be drawn.
The so-called nearly linear input-output relationships of non-touch mode capacitive pressure sensors can be easily realized by using the presented analytical solution-based method. It can be seen from Section 3 that the desired nearly linear input-output relationships can be easily achieved by changing design parameters, such as membrane thickness, Young's modulus of elasticity and the initial gap between the initially flat undeflected conductive membrane and the insulator layer coating on the substrate electrode plate. Therefore, the view in the literature is open to debate that non-touch mode capacitive pressure sensors are far inferior to touch mode capacitive pressure sensors in the easy realization of nearly linear input-output relationships.
The change in membrane thickness has no effect on the range of input capacitance and only affects the range of output pressure, which increases with the increase in membrane thickness.
The change in Young's modulus of elasticity affects both the range of output pressure and the range of input capacitance, where the range of output pressure increases with the decrease in Young's modulus of elasticity, and the range of input capacitance decreases with the decrease in Young's modulus of elasticity.
The change in Poisson's ratio has a very limited effect on input-output relationships. Therefore, it is sufficient to know the approximate range of Poisson's ratio rather than its exact value when choosing a polymer conductive membrane as the movable electrode plate of a capacitor of a non-touch mode circular capacitive pressure sensor.
The change in insulator layer thickness has no effect on the range of output pressure and only affects the range of input capacitance, which decreases with the increase in insulator layer thickness.
in the plane in which the geometric middle plane of the initially flat circular membrane is located). Let us take a free body with radius 0 ≤ r ≤ a from the deflected circular membrane under uniformly distributed transverse loads q, as shown in Figure A2, to study its static problem of equilibrium.

Appendix A
A peripherally fixed, initially flat and taut linearly elastic circular membrane with Young's modulus of elasticity E, Poisson's ratio ν, thickness h, and radius a is subjected to a uniformly distributed transverse loads q, as shown in Figure A1, where r is the radial coordinate, w is the transversal displacement, o is and the original point of the introduced cylindrical coordinates system (r, ϕ, w) (where the polar coordinate plane (r, ϕ) is located in the plane in which the geometric middle plane of the initially flat circular membrane is located). Let us take a free body with radius 0 ≤ r ≤ a from the deflected circular membrane under uniformly distributed transverse loads q, as shown in Figure A2, to study its static problem of equilibrium.  In the vertical direction perpendicular to the initially flat circular membrane, there are two vertical forces acting the free body, that is, the πr 2 q produced by the loads q within r, and the 2πrσrhsinθ produced by the membrane force σrh, where σr is radial stress. So, the out-of-plane equilibrium condition is

Appendix A
A peripherally fixed, initially flat and taut linearly elastic circular membrane with Young's modulus of elasticity E, Poisson's ratio ν, thickness h, and radius a is subjected to a uniformly distributed transverse loads q, as shown in Figure A1, where r is the radial coordinate, w is the transversal displacement, o is and the original point of the introduced cylindrical coordinates system (r, ϕ, w) (where the polar coordinate plane (r, ϕ) is located in the plane in which the geometric middle plane of the initially flat circular membrane is located). Let us take a free body with radius 0 ≤ r ≤ a from the deflected circular membrane under uniformly distributed transverse loads q, as shown in Figure A2, to study its static problem of equilibrium.  In the vertical direction perpendicular to the initially flat circular membrane, there are two vertical forces acting the free body, that is, the πr 2 q produced by the loads q within r, and the 2πrσrhsinθ produced by the membrane force σrh, where σr is radial stress. So, the out-of-plane equilibrium condition is In the vertical direction perpendicular to the initially flat circular membrane, there are two vertical forces acting the free body, that is, the πr 2 q produced by the loads q within r, and the 2πrσ r hsinθ produced by the membrane force σ r h, where σ r is radial stress. So, the out-of-plane equilibrium condition is 2πrσ r h sin θ = πr 2 q, where sin θ = 1/ 1 + 1/ tan 2 θ = 1/ 1 + 1/(−dw/dr) 2 .
While in the direction parallel to the initially flat circular membrane, the equilibrium condition may be written as [36] d(rσ r ) dr where σ t denotes circumferential stress. The derivation of Equation (A4) is detailed in [36]. If the radial and circumferential strain and the radial displacement are denoted by e r , e t and u, respectively, then the relationships between strain and displacement for large deflection problems may be written as [37] Moreover, the relationships between stress and strain are still assumed to satisfy linear elasticity and expressed in terms of generalized Hooke's law [38] σ r = E 1 − ν 2 (e r + νe t ) (A7) and σ t = E 1 − ν 2 (e t + νe r ). (A8) Substituting Equations (A5) and (A6) into Equations (A7) and (A8) yields and By means of Equations (A4), (A9) and (A10), one has After substituting the u in Equation (A11) into Equation (A9), we obtain an equation containing only the radial stress σ r and deflection w(r) Equations (A3) and (A12) are two equations for solving the radial stress σ r and deflection w(r). The boundary conditions, under which the particular solutions of the radial stress σ r and deflection w(r) can be determined, are w = 0 at r = a, (A13) u = 0 at r = a (A14) and dw dr = 0 at r = 0.