Characterization of Capacitive Comb-finger MEMS Accelerometers

This paper discusses various methods for testing the performance of MEMS capacitive comb-finger accelerometers manufactured by Sandia National Laboratories. The use of Capacitive MEMS devices requires complex circuits for measurement of capacitance. Sandia MEMS accelerometer's capacitance changes in a very small femto-farad (fF) range. The performance of accelerometer is tested using Analog Devices AD7747 sigma-delta capacitance to digital converter. The response of a MEMS capacitive accelerometer to various tests is useful for testing and characterization and investigate it's suitability for various applications


Introduction
In the past two decades, various capacitive MEMS accelerometers have been widely commercialized. Capacitive sensing is one of the most popular sensing mechanisms of inertial sensors because it has the advantages such as low power consumption, low noise, low-cost and compatibility with IC fabrication technology. However, the effect of parasitic capacitance is one of the main concerns for this type of capacitive sensors. A capacitive comb-finger accelerometer has a series of interlocking fixed and movable fingers, which are attached to the proof mass to measure the specific force through a change in capacitance. When the external acceleration is applied, the gap between the fixed and moving fingers changes, which corresponds to change in differential capacitance. This change can be measured using a suitable electronic circuitry [1,2]. The Sandia MEMS accelerometers are manufactured using CMOS SUMMiT V fabrication process. The devices fabricated using SUMMiT V are truly complex and more advance systems can be created using this technology [3]. The MEMS accelerometer is capable of sensing change in capacitance in femto-farad range. In this paper, change in capacitance is measured using the Analog Devices AD7747 evaluation board. The AD7747 is sigma delta capacitance to digital converter, which converts capacitive input signal into digital values. This capacitance to digital converter is specifically designed for capacitive sensors [4].

Sandia Laboratories Capacitive MEMS Accelerometer
The Sandia laboratories MEMS accelerometers are manufactured using CMOS SUMMiT V fabrication process. It is a five-layer polycrystalline silicon surface micromachining process With the SUMMiT V process; more advanced, complex and flexible systems can be created at low-cost. Figure 1 shows the magnified view of a Sandia comb-finger accelerometer structure [3].

Mechanical Sensitivity of Sandia Accelerometer
MEMS accelerometer sensitivity is an important design parameter. Sensitivity is the ratio of change in electrical signal (output) to change in physical signal (input). A spring mass model is used to model physical motion of accelerometers. For Sandia accelerometer, total sensitivity is a product of mechanical sensitivity and electrical sensitivity.
Mechanical sensitivity has the unit of displacement per unit gravitational acceleration (meters/g). Mechanical modeling can be studied by understanding spring mass model for a genericaccelerometer as shown in Figure 2 [5,6].  Figure 3 shows four beam folding structure of Sandia device. The accelerometer silicon proof mass is designated "m" and is suspended by a spring "k" to a frame. The spring operation follows the Hooke's law, which contains the following relationship between the spring displacement and the force F [5,6]; Where k is the spring constant and is the displacement of the mass (m). From Newton's law,  Figure 3. Sandia Capacitive MEMS accelerometer structure [6] The beam springs, which are fixed at one end behave very much like a double cantilever beam. Figure [  . Cantilever beam movement with one rolling beam end [5] For this particular geometry of the comb finger accelerometer, the spring constant is calculated using Eq. (4) (4) where is the moment of inertia of the beam:

Bulletin of EEI
The entire accelerometer would have four such folds, thus we have total spring constant of the whole accelerometer is calculated using Eq. The system can be redrawn with spring mass system model as shown in Figure 2. Using Eq. (3) mechanical sensitivity can then be calculated when 1 g (9.8 / )is applied to system. * where m is the value of accelerometer proof mass and g is gravitational acceleration. K system is the spring constant or stiffness for the system [6].

Electrical Sensitivity of Sandia Accelerometer
The equation of capacitance shown below is dependent on the displacement between the two plates, which is the basis for using capacitive comb-finger accelerometer thus it is sometimes called electrostatic accelerometer [5].
Where ε r is the dielectric constant between the plates, ε o is the electric constant, A is the area of the two plates and d is the distance between the two plates.
Figure 5 [5] shows the proof (shuttle) mass sitting between the fixed fingers with no displacement. is the capacitance of left side and is the capacitance of right side.

Figure 5. Capacitive interaction between movable and fixed fingers [5]
When there is no acceleration then differential capacitance is zero but when proof mass moves in the direction of acceleration, one capacitance will be larger than the other resulting in non-zero reading of the differential capacitance as shown in Figure 6 and Figure 7 [5]. Each Capacitor can be seen as capacitive reactance so the voltage difference equates the difference between the voltage drops across each capacitor. The magnitude of the voltage difference is proportional to the magnitude of the capacitance change [5] Bulletin of EEI The electrical modeling involves measuring the difference in capacitance to be simplified by creating all electrodes with the same area. and can be used to determine the output of the accelerometer circuit. For testing point of view, the accelerometer can be seen electrically as a capacitive voltage divider as shown in Figure 8 [1,7]. For this circuit, The electrical sensitivity of the comb-finger incorporates both types of electrode movements lateral and transverse as shown in Figure 9 and Figure10 [6]. Electrical sensitivity is the change in capacitance is due to change in displacement. The electrical sensitivity due to lateral movement is given in Eq. (11),

Bulletin of EEI
The total electrical sensitivity has to take into account the plate separation so equation for total electrical sensitivity is given in Eq. (12), The second term in [Eq.13] which is plate separation term in can be neglected if ∆x<<d o and because d 02 is a large value compared to d 01 and it is squared which makes the second term small compared to the first term. The total electrical sensitivity for Sandia comb-finger accelerometer is calculated as follows [5,6]; (13) 2 8.86e 12 F/m 1080 ingers 4.5e 6 m 1.25e 6 m =68.9 fF/μm (14)

Total Sensitivity
Total sensitivity (S T ) of the Sandia accelerometer is the product of mechanical sensitivity (S M ) and electrical sensitivity (S E ) [5,6].
To prevent the device from damage it was necessary to know the range of the accelerometer before performing any test. Mechanical and electrical sensitivity of Sandia accelerometer is known. These parameter values are mentioned in Table 1. The range of accelerometer can be found using Eq. 16 and Eq. 17.
. / = 60.6796 g But for the factor of safety actual range of Sandia accelerometer is considered as, * 60.6796 6 (17)

Analog Devices AD7747 Evaluation Board
AD7747 is a high resolution, 24 bit sigma delta capacitance to digital converter. It is designed for single-ended or differential capacitive sensors with one plate connected to ground. It also has tolerance of parasitic capacitance to ground up to 60pF. The capacitive accelerometer input is connected to the inputs on a board and the required operational settings can be loaded through evaluation board and supported software though computer interfacing. The AD7747 core as shown in Figure 10 is a high precision converter consisting of a secondorder sigma-delta modulator and a third order digital filter. It works as a CDC (Capacitance to Digital Converter) for the capacitive inputs and ADC (Analog to Digital Converter) for the voltage input from or for the voltage from a temperature sensor. With the converter AD7747 also integrates multiplexer, an excitation source and CAPDACs (Digital to Capacitance converter) for the capacitive input. AD7747 supports I C compatible serial interface, a control and calibration logic. AD7747 is not factory calibrated for capacitive offset so its user's responsibility to calibrate the system capacitive offset in the application [4,9].

Test and Measurement
It is necessary to check the packaged MEMS devices before actual testing.

Continuity Test
A primary method of testing the MEMS device is to check the discontinuity between fingers and different layers. This test provides information about proper structure of fingers and good devices can be easily sorted out.

LCR Meter Test
Testing a damaged unit could result in inaccurate data so to quantify viable test accelerometer without any risk was possible with a LCR meter. (Inductance (L), Capacitance (C) and Resistance (R)). As Sandia accelerometers should read very small capacitance with very high resistance/impedance, it is assumed that if both the sides of device read small capacitance with high resistance then it is good for further testing. The reasoning here is that if the proof mass and comb-fingers are connected on the inside, it will lose its capacitance and it will show drop in resistance over that side.

Tip Test
The simple way to see if capacitive accelerometer is sensing a difference in capacitance is a Tip Test. To perform this test first capacitance is measured when accelerometer is at rest, then it is positioned in such a way that its sensitive axis is perpendicular to the earth gravitation vector. The difference in acceleration can be observed and capacitance measurement is done AD7747. The test setup is shown in Figure 11 (when accelerometer is at '0' g acceleration) and 12 (when accelerometer is at '+1g' and '-1g' acceleration), respectively.
Offset is nothing but nulling zero error or bias and scale factor or sensitivity of an accelerometer is the ratio of the sensor electrical output to mechanical input typically rated in

Thermal Test
The thermal test is designed to determine the thermal stability characteristics of accelerometer at particular temperature range. To avoid the risk of damage temperature was slowly increased from for this test. Figure 11 shows thermal test result for Sandia MEMS accelerometer. The trend of decreasing capacitance with increase in temperature is observed from the test result.

Hysteresis Test
A sensor (accelerometer) should be capable of following changes in capacitance (input parameter) regardless of which direction the change is made and hysteresis is the measure of this property. To perform this test temperature was slowly increased and then decreased from room temperature to the highest temperature which device could sustain. Hysteresis is expressed as % of FSO. The hysteresis plot is shown in Figure 15. From Figure 15

Vibration Test
To conduct vibration test, accelerometer is mounted on shaker table and acceleration is increased and change in capacitance is observed at different acceleration values. To avoid device damage, vibration test is performed at 0.5g, 1g and 1.5 g accelerations. The vibration test set up is shown in Figure 16 and test results are shown in Figure 17.

Noise Analysis using Allan Deviation
Allan deviation is a statistical analysis method that can be used to measure noise error and bias stability in inertial sensors. It can identify and measure the magnitude of each type of noise error in a signal of mixed noise. The slope at different points in the plotted Allan deviation graph determines the different noise errors. The MEMS accelerometer noise typically consists of the following terms, which can be represented using Allan variance plots [10].  Quantization noise: It is one of the errors introduced into an analog signal by encoding it in digital form. That noise is caused by the small differences betweenthe actual amplitudes of the points being sampled and the bit resolution of the analog-to-digital converter. The quantization noise is represented by a slope of -1 in a log-log versus .  Angular random walk: This is an angular error process, which is due to white noise in angular rate. ARW is a high frequency noise and it can be observed as the short-term variation in the output. It is represented by a slope -1/2 in log-log versus .  Bias instability: The bias stability is also known as flicker noise. This is a low frequency bias fluctuation in the measured data. Bias instability determines the best stability that could be achieved with fully modeled sensor. Figure 18 to Figure 20 shows Allan deviation, Quantization noise and Angular random walk plots for Sandia 14 pin accelerometer. The ARW can be determined from this plot as the Allan deviation corresponding to = 1 sec, is approximately 4.1E-6. The bias stability of the accelerometer is the minimum Allan deviation approximately 0.2E-6.

Conclusions
In this paper, exhaustive methods of testing micro-machined capacitive comb-finger accelerometer are discussed. The performance of Sandia capacitive comb-finger MEMS accelerometer is studied under various conditions. The change in capacitance is measured by using an Analog devices AD7747 capacitance to digital converter. This framework and process of analysis and testing using various design parameters is helpful for characterization of any MEMS capacitive sensor. The major findings of this research are mentioned in Table 2.