1. Introduction

The micro-silicon accelerometer based on MEMS technology has been considered as an attractive choice in many application areas for its small size, light weight, and high sensitivity, along with its strong anti-radiation property, ease of mass-production and other features. Especially as a kind of inertial device, it has been widely used in the field of aerospace, automatic control, vibration testing, biology, chemistry, medical analysis, etc^[1-3]. Because of the different sensitive mechanisms, the accelerometers can be divided into various forms such as piezoresistive^[4], piezoelectric^[5], capacitive^[6] and so on. Accelerometers based on different principles have their own advantages and defects.

Although the capability of piezoresistive accelerometers is greatly influenced by the change of temperature, the advantages that simple structure and production process, good DC response, high reliability and low cost provide a potential possibility for its batch production^[7-9]. The piezoelectric accelerometer has characteristics of low cost, simple structure, ease of integration, wide band, high sensitivity and fast response, but its disadvantages such as a poor low-frequency response and no signal for constant acceleration, cannot be ignored^{[5, 10, 11]}. The advantages of the capacitive accelerometer are low drift, low temperature sensitivity and having DC response, but the signal detection is very difficult due to the existence of parasitic capacitance^[12]. Compared with the piezoelectric accelerometer and capacitive accelerometer, the sensitivity of a piezoresistive accelerometer is not so high in normal conditions. Nevertheless, the piezoresistive accelerometer still has a lot of appeal with its simple structure, high sensitivity, small non-linearity, low power dissipation, etc. Usually there are two methods to improve the sensitivity: one uses a high-strain coefficient material^[13], the other is by structure innovation^[14]. This paper introduces a kind of detection method named dual cycle bridge detection, which is based on the accelerometer of an "eight-beams/mass" structure. All the 16 resistors in the eight beams are divided into two groups to form a cycle detection bridge respectively, clearly eight resistors in each bridge form a cycle detection bridge (No. 1 and No. 2). The final output signal is the additive results of the two cycle detection bridges on the $x, y, z$ directions. The rationality of the dual cycle bridge detection has been analyzed here. By modeling, simulating, calculating and comparing with the common independent Wheatstone bridge detection method, it has been found that the method of dual cycle bridge detection can improve the sensitivity of the accelerometer effectively. The practical results and the theoretical analysis results are consistent on the whole by testing the accelerometer on the precision centrifuge.

2. The design of the accelerometer

A 3D model of the accelerometer is shown in Fig. 1. The structure is designed as "eight-beams/mass", which is composed of the surrounding frame, beams and mass. Each two of the eight beams are designed to be suspended between the mass and the frame symmetrically, and the resistors are formed by ion implantation as the sensitive device. Figure 2 shows the symmetrical distribution of the beams. The 16 resistors, $R_{{\rm X}ij}$, $R_{{\rm Y}ij}$($i$= 1, 2; $j$= 1, 2, 3, 4), are divided into two groups with two colors and eight resistors in each group constituting a cycle detection bridge.

2.1 Principle of the cycle bridge detection

When applying acceleration load on the accelerometer, the center mass will produce a displacement due to the effect of inertia force. The movement of the mass leads to the deformation of beams. According to the mechanics of materials, the stress on the surface at both ends of the beam is directly opposite. That is to say, when one of the resistors on the beam bears tension stress, the other will undertake compressive stress. At the same time, the resistance value of the resistors will also change oppositely for the difference stress. In order to reduce the cross-axis coupling degree of the triaxial accelerometer, it should be ensured that the resistance value variation on the beam is the same. So the right location of the resistor is important to a certain extent. The output voltage value of the cycle detection bridge reflects the acceleration loaded on the accelerometer. Table 1 shows the circuit diagrams, the corresponding voltage output expressions, as well as the final voltage output expressions of the structure in Fig. 2.

When a 1$g$ acceleration load is applied in the $x$ direction of the accelerometer, the centre mass displacement will cause the four beams to bend in this direction, which leads to a resistance value variation $\Delta R_{1}$ of the eight beams: the ones under tension will decrease $\Delta R_{1}$ while the others oppositely increase $\Delta R_{1}$ under the pressure. By the effect of the centre mass displacement, the four beams in the $y$ direction will warp and the eight resistors will change $\Delta R_{2}$ ($\Delta R_{2}$ $\ll$ $\Delta R_{1})$. The structure deformation of the accelerometer is shown in Fig. 3(a) and the resistors change of the beams is shown in Fig. 3(b). According to Eqs. (1)-(13), the output voltage of the cycle detection bridge can be figured out and the result is shown in Table 2.

Since the structure of the accelerometer is symmetrical, the result of applying 1$g$ acceleration load in the $y$ direction is similar to the one in the $x$ direction. The structure deformation of the accelerometer is shown in Fig. 4(a) and the resistors change of the beams is shown in Fig. 4(b). The output voltage of the cycle detection bridge is shown in Table 2.

When applying a 1$g$ acceleration load in the $z$ direction of the accelerometer, the centre mass moves along the $z$ direction and the eight beams will bend under the same force effect. When the mass moves down, the resistors of the eight beams near the mass will increase $\Delta R_{1}$ under the pressure while the resistors of the other eight ones near the frame will decrease $\Delta R_{1}$ under the effect of tension. When the mass moves up, the force situation is just the opposite. The structure deformation of the accelerometer is shown in Fig. 5(a) and the resistors change of the beams is shown in Fig. 5(b). The result of the output voltage of the cycle detection bridge is shown in Table 2.

It can be known from the above results and analysis that for the triaxial accelerometer of an "eight-beams/mass" structure, with proper design, the detection for output signals in three directions can be completed by each cycle detection bridge (No. 1 or No. 2), which is composed of eight of the 16 beams. Furthermore, if all 16 resistors are divided into two groups and each group constitutes a cycle detection bridge to provide real-time detection of the output signal in the $x, y$, and $z$ directions. The value of the final output voltage obtained by adding the output signals of two cycle detection bridges will be twice as much as one bridge. So the conclusion can be drawn that the dual cycle bridge direction is feasible for the detection of a triaxial accelerometer. In order to prove the superiority of this method on improving the sensitivity, the independent Wheatstone bridge detection method is analyzed as follows (the premise is the same sensor). The circuit diagrams of independent Wheatstone bridges in the $x, y,$ and $z$ directions are shown in Fig. 6. The corresponding output voltage can be obtained according to Figs. 3(a), 4(a), 5(a) and Eqs. (14)-(16), then the results are summarized in Table 3.

$V_{\rm out} X=\left( {\frac{R_{\rm X13} }{R_{\rm X11} +R_{\rm X13} }-\frac{R_{\rm X14} }{R_{\rm X12} +R_{\rm X14} }} \right)V_{\rm CC},$

(14)

$V_{\rm out} Y=\left( {\frac{R_{\rm Y12} }{R_{\rm Y12} +R_{\rm Y14} }-\frac{R_{\rm Y11} }{R_{\rm Y11} +R_{\rm Y13} }} \right)V_{\rm CC},$

(15)

$\begin{split} \label{eq3} V_{\rm out} Z ={}& \left( \frac{R_{\rm X21} +R_{\rm X22} }{R_{\rm Y23} +R_{\rm Y24} +R_{\rm X21} +R_{\rm X22} }\right. \\[2mm]& \left. -\frac{R_{\rm X23} +R_{\rm X24} }{R_{\rm Y21} +R_{\rm Y22} +R_{\rm X23} +R_{\rm X24} } \right)V_{\rm CC} . \end{split}$

(16)

By comparing the result in Tables 2 and 3, it can be found that the dual cycle bridge detection is better than the independent Wheatstone bridge detection method on improving the sensitivity of the triaxial accelerometer. Specifically, in the $x$ and $y$ directions, the sensitivity is twice as much as the independent Wheatstone bridge detection method, and four times as much as in the $z$ direction.

2.2 The finite element simulation

For further verification of the superiority on improving the sensitivity of this method, ANSYS is used for modeling and analyzing. The simulation diagrams of structure deformation with respective load in the $x, y$, and $z$ directions of the accelerometer are shown in Fig. 7. Compared with the theory analysis on Figs. 3(a), 4(a) and 5(a), the results are consistent on the whole. By defining the path of one of the beams, extracting and analyzing the path data, the location can be determined where the resistance value variation of the two beams is equal.

As a sensitive element, the resistors greatly affect the accelerometer's performance. The sensitivity of the accelerometer is not only affected by the location of resistors, but also related to the type of resistors. Since P-type resistors have a larger resistance change rate than the N-type, P-type resistors will be a better choice. The change rate of the resistor is^[15-17]:

$\frac{\Delta R}{R}=\pi _{\rm l} \sigma _{\rm l} +\pi _{\rm t} \sigma _{\rm t} +\pi _{\rm s} \sigma _{\rm s} .$

(17)

In the formula, $\sigma _{\rm l}$ is transverse stress, $\sigma _{\rm t}$ is longitudinal stress, $\sigma _{\rm s}$ is vertical stress, $\sigma _{\rm s}$ is generally ignored as it is far smaller than $\sigma _{\rm l}$ and $\sigma _{\rm t}$; $\pi _{\rm l}$, $\pi _{\rm t}$, $\pi _{\rm s}$ are piezoresistance coefficients respectively corresponding to $\sigma _{\rm l}$, $\sigma _{\rm t}$, $\sigma _{\rm s}$. So Equation (17) is simplified as:

$\frac{\Delta R}{R}=\pi _{\rm l} \sigma _{\rm l} +\pi _{\rm t} \sigma _{\rm t} =\left( {71.8\sigma _{\rm l} -66.3\sigma _{\rm t} } \right)\times 10^{-11}.$

(18)

By extracting the force data of the resistors on the beams and substituting it into Eq. (18), the resistance change rate can be obtained. According to Eqs. (1)-(13) in Table 1, the output voltage of the cycle detection bridge can be figured out. Convert this output voltage to the one under a 1$g$ acceleration load, the sensitivity of the accelerometer will be eventually obtained (the ratio of the output voltage and acceleration load is the sensitivity). The simulation result of the accelerometer sensitivity and cross-axis coupling degree is shown in Table 4.

From the result in Table 4, it can be found that the detection for output signals in three directions can be completed by one cycle detection bridge (No. 1 or No. 2). Furthermore, the sensitivity in the $x$ and $y$ directions is almost the same, while in the $z$ direction, it is 8.8 times as much as the one in the $x$ or $y$ direction. The dual cycle bridge detection has the advantages that not only improve the sensitivity but also reduce the cross-axis coupling degree by counterbalancing the effect of lateral output.

3. The design of the detection circuit

Even though the dual cycle bridge detection can improve the sensitivity, the output voltage of the triaxial accelerometer based on MEMS technology is still too small for analysis. So an amplification process for the output signals is necessary for obtaining an accurate value of the acceleration in the three directions. Besides, in the dual cycle bridge detection, adding the output signals is required. So a simple amplifier circuit and adder circuit for the output signals are provided here for the detection process.

3.1 The design of the amplifier circuit

Since the output signal of the accelerometer is voltage, the AD620 precision instrumental magnifier is chosen for its low cost, low noise and high precision. The amplifier circuit diagram is shown in Fig. 8(a) and Figure 8(b) is the simulation result after amplification.

From Fig. 8(b), the actual magnification factor can be figured out as 50.46, which is very close to the theoretical amplified value 50.4. That is to say, the amplifier circuit can meet the requirements of precise amplification.

3.2 The design of the adder circuit

As LM224 is an operational amplifier with low cost and wide power supply voltage, it is used for connecting to the inverting adder circuit for adding the signal in the detection circuit. Figure 9(a) shows the adder circuit diagram and Figure 9(b) is the simulation result after the inverting addition.

From Fig. 9(b) it can be found that the result of directly adding the signals is the same as the one of adding by an inverting adder circuit. That is to say, the adder circuit meets the requirements of the precise addition. However, the resistance selection is complicated in the actual debugging process for meeting the requirements of $R_{2}=R_{3}=R_{6}$ and $R_{1}=R_{2}/\!/R_{3}/\!/R_{6}$.

4. The results and discussions of the test

The fabrication of an accelerometer includes resistors diffusion, the mass manufacture and etching of eight beams. Due to that, SOI wafers can accurately control the thickness of the device, and N-type silicon SOI wafers are used for the fabrication of the device. Resistors are made by diffusing B$^+$ on n-type silicon, while the mass and beams are made by ICP etching. The SEM photos and packaging photo of the designed triaxial accelerometer are shown in Fig. 10.

It is relatively difficult to test the accelerometer, especially for the low-$g$ accelerometer, as any deviation of installation can lead to measurement errors for its high sensitivity. A precision centrifuge provides a reliable method to test the low-$g$ accelerometer as it has stable rotation and an accurate output. The accelerometer is installed in the rotating platform of the precision centrifuge, where one must ensure the measured direction is pointing to the axis of the precision centrifuge. The rotating platform of the precision centrifuge is balanced by a same-mass accelerometer. A constant acceleration is loaded on the accelerometer in the axis direction of the precision centrifuge when the rotating platform revolves at a constant speed. Then the values of the output voltage in three directions will respectively be displayed on the screen of the digital multimeter. The output voltage in the $x$ and $y$ directions is magnified 20 times and 5 times in the $z$ direction for a more accurate test data. The test process is shown in Fig. 11. The test results of No. 1 and No. 2 single cycle detection bridges are shown in Figs. 12 and 13, and the result of the dual cycle detection bridge is shown in Fig. 14.

Comparing the above test results with the simulation results in Table 4, there is some bias between the actual sensitivity and the simulated of the accelerometer in the $x$, $y$, and $z$ directions. To be specific, firstly, the actual sensitivity is lower than the simulation, which may be caused by the actual beam being thicker compared with the simulated one. Secondly, the sensitivity in the $x$ and $y$ directions should be the same in theory, while the actual result shows a difference. Meanwhile, no matter whether the No. 1 or No. 2 bridge is used, the sensitivity in the $x$ direction is higher than in the $y$ direction and the coupling degree of $x$ to $y$ is lower than $y$ to $x$. It could be argued that the cause of the results may be an error of installation, that the measured direction did not accurately point to the axis of the precision centrifuge when the sensitivity in the y direction was tested. For the bias of the sensitivity in the $z$ direction, the main cause may come from the fabrication process, which includes the fabrication of the accelerometer's structure and the consistency of the resistance of the resistors. Also, the bias in the fabrication process will be an irresistible cause for all the above errors.

It can be found from the above test result that the dual cycle detection bridge not only improves the sensitivity but also reduces the cross-axis coupling degree. For example, from the No.2 cycle detection bridge test result, the coupling degree of $x$ to $z$ is 5.01% and $y$ to $z$ is 2.28%. While with the dual cycle detection bridge, the coupling degrees respectively reduce to 1.72% and 1.43%. For other directions, the dual detection bridge is also effective in decreasing the coupling degree compared with the single cycle detection bridge by counterbalancing the effect of lateral output. Specifically, the cross-axis coupling degree of the dual cycle detection bridge is between No. 1 and No. 2.

5. Conclusion

This paper presents a dual cycle detection bridge specific to the piezoresistive triaxial accelerometer based on MEMS technology. In order to verify the feasibility of this method, the dual cycle detection bridge and independent Wheatstone bridge detection methods are compared in the detection of the triaxial accelerometer of an "eight-beams/mass" structure. Through theory analysis, it is known that the cycle bridge detection method can improve the sensitivity effectively. In order to estimate the sensitivity of the accelerometer that used the cycle bridge detection method, ANSYS is used to simulate, and then calculate the sensitivity of the sensor. By testing the manufactured triaxial accelerometer, the sensitivity results of the dual cycle detection bridge are 113.23 $\mu$V/g in the $x$ direction, 108.91 $\mu$V/g in the $y$ direction and 1010.08 $\mu$V/g in the $z$ direction. The cross-axis coupling degree is lower than 2%. The dual cycle detection bridge not only improves the sensitivity but also reduces the cross-axis coupling degree by counterbalancing the effect of lateral output.