#### 2.1. Influence of Parasitic Capacitance

Figure 1 shows the schematic of effective and parasitic capacitances in MEMS capacitive accelerometer interfaced with a C/V converting circuit. Obviously, there are several parasitic capacitances and the mismatch of parasitic capacitances ΔC

_{m1} between C

_{p1} and C

_{p2}—including in the sensor, package, and circuit—will confuse the differential effective capacitances ΔC between C

_{top} and C

_{bottom} that would produce an offset. The mismatch ΔC

_{m2} between C

_{p3} and C

_{p4} will also have an influence on the output. Besides, the parasitic capacitances, C

_{p5} and C

_{p6}, can affect the influence of ΔC

_{m1} and ΔC

_{m2} on the output.

Generally, the sensitivity of effective capacitance is about 100 fF/g or even smaller and the mismatch of parasitic capacitance can be up to 100 fF that will result in an offset of 1 g. This large offset would severely deteriorate the performance of the accelerometer. Therefore, it is necessary to study the mismatch and do some work for reducing the influence. Measuring the mismatch accurately is a basic step. Though there are many discrete parasitic capacitances, we only need to obtain the total equivalent mismatch.

#### 2.2. Theory of Measuring the Mismatch

In the closed-loop system of a MEMS capacitance accelerometer, there is electrostatic force between fixed plates and proof mass that balances the inertial force caused by acceleration [

13], and the proof mass is not at the geometrical center for the mismatch of parasitic capacitance.

Figure 2 shows a working diagram of the sensor.

Considering the process variation and parasitic capacitance, the electrostatic force

${F}_{e}$ of the proof mass is:

where

${\epsilon}_{r}$ and

${\epsilon}_{0}$ are the relative and absolute dielectric constant respectively,

$A$ is the overlapped area of capacitance,

${V}_{d}$ is the modulated voltage,

${V}_{fb}$ is the feedback voltage,

${V}_{ref}$ is the pre-load voltage,

${d}_{0}$ is the average gap between electrodes,

$\Delta d$ is the gap deviation due to process variation, and

$x$ is the bending value of the beam due to the mismatch of effective and parasitic capacitance.

In general,

$x$ and

$\Delta d$ are far smaller than

${d}_{0}$, and then, Equation (1) can be simplified to:

where the bending value

$x$ consists of

${x}_{1}$ brought by the mismatch of effective capacitance and

${x}_{2}$ brought by the mismatch of parasitic capacitance, so

$x={x}_{1}+{x}_{2}=-\Delta d+{x}_{2}$. Substituting this equation to Equation (2), the electrostatic force

${F}_{e}$ can be expressed as:

where

$2{\epsilon}_{r}{\epsilon}_{0}A\times \left({V}_{ref}^{2}+{V}_{fb}^{2}+{V}_{d}^{2}\right)/{d}_{0}^{3}={k}_{e}$ is called electrostatic stiffness.

In the closed-loop system, there is the force balance for the proof mass:

where

$k$ is the stiffness of the beam,

$m$ is the inertial mass of the proof mass,

$a$ is the external acceleration, and

${F}_{s}$ is the residual stress. Replacing Equation (3) into Equation (4), the formula of force balance can be expressed as:

where

${B}_{0}=2{\epsilon}_{r}{\epsilon}_{0}A\times {V}_{d}^{2}\times {x}_{2}/{d}_{0}^{3}-kx-ma-{F}_{s}$. When the input acceleration is unchanged, the parameter

${B}_{0}$ can be considered as a fixed value. When the input acceleration and offset are small,

${V}_{fb}^{2}$ is far smaller than

${V}_{ref}^{2}$, so Equation (5) can be simplified to:

For the digital acquisition system, the left portion in Equation (6) can be transformed to

${F}_{e}^{\prime}=2{\epsilon}_{r}{\epsilon}_{0}A\times {V}_{ref}{V}_{fb}/{d}_{0}^{2}={U}_{out}/{K}_{1}\times m\times {g}_{L}$ where

${U}_{out}$ is digital output which unit is LSB,

${K}_{1}$ is the scale of accelerometer which unit is LSB/g and

${g}_{L}$ is local gravity acceleration. Then, Equation (6) can be transformed to:

Equation (7) can be transformed to:

where

$Y={U}_{out}/{K}_{1}\times m\times {g}_{L}$ is dependent variable,

$X={V}_{ref}^{2}$ is independent variable,

${B}_{1}=2{\epsilon}_{r}{\epsilon}_{0}A\times {x}_{2}/{d}_{0}^{3}$ is linear coefficient and

${B}_{0}$ is intercept which is a fixed value.

Equation (8) shows that the relationship between output of electrostatic force

${F}_{e}^{\prime}={U}_{out}/{K}_{1}\times m\times {g}_{L}$ and the square of pre-load voltage

${V}_{ref}^{2}$ is linear. Thus, we can make a curve with

${F}_{e}^{\prime}$ as

y-axis and

${V}_{ref}^{2}$ as

x-axis, and then, a linear fitting of the curve is made. Lastly, the mismatch of the parasitic capacitance can be obtained from the linear coefficient

${B}_{1}$ through the equation:

where

${d}_{0}$ can be calculated through the obtained scale of the closed-loop system. Meanwhile, we can get the offset and the deviation from geometrical center due to the mismatch of parasitic capacitance.