https://www.ericcointern...flex-accelerometers.html 

Aerospace-Quartz-Accelerometer-3
At the heart of Q-Flex accelerometers technology lies a delicate balance of mechanical and electrical components, each meticulously designed to interact seamlessly within the device's closed-loop system. Yet, variations in temperature can introduce subtle changes in material properties, altering the device's sensitivity and response characteristics. This article proposes a novel temperature compensation procedure, which can simultaneously reduce the negative effect brought by temperature and roll angle.The proposed method is composed of the following steps.



1.Modeling Temperature Curves of Maximum and Minimum.



The bias and scale factor may suffer deteriorated performance when the Q-Flex accelerometer is used in high temperature and high accuracy required measurement system, probably because of different nonlinear temperature drift of the complex peripheral interface circuit. Therefore, for better expressing the temperature drift, we adopt a polynomial model here to obtain the temperature curve of maximum and minimum output of the sensor, which can be written as



Equation 1.



where gm is the maximum or minimum output, αi is the corresponding nonlinear coefficient, and t is the temperature. n is empirically set to 4 in our system due to the trade off between the effective repeatability of the maximum and minimum output and overfitting. In fact, the sensor is not sensitive at the position of the maximum or minimum output, so it is easy to find the proper position for maximum or minimum output in the actual operation (for convenience, the compensated Q-Flex accelerometer is mounted into a cylinder structure, and the direction of the maximum sensor output is the same with the gravity).



2.Modeling Temperature Curve of Zero Sensor Output.



The key step, also the significant difference of our proposed procedure, is to obtain the temperature curve of the zero sensor output. Due to the manufacturing technique or internal structure, the temperature curve of the zero sensor output changes with the roll angle at a fixed inclination, which makes the usual temperature compensation method less effective. The direct result is that the normalized positive and negative parts are asymmetrical at high temperature. Formally, that is, (gout − g0)/(gmax − g0) ≠ (gout − g0)/(g0 − gmin), where gout is the actual output of the sensor and gmax, gmin, and g0 are maximum, minimum, and zero sensor output, respectively. To handle this problem, we need to obtain a more accurate temperature curve of zero sensor output.



Assume that the function between the zero sensor output and the temperature is linear (Assumption 1), which could be defined as



Equation 2.



where k is the slope and b is the offset. Suppose that two points (t1, y1) and (t2, y2) can determine this linear function, in which (t1, y1) is a point at room temperature t1, so k could be calculated by k = (y2−y1)/(y2−y1) and b could be b=y1−k·t1. According to Assumption 2, if we set Δy = y2 − y1, then Δy could satisfy the function



Equation 3.



where A is the amplitude, ω·θroll + φ is the phase, and ℎ is the offset. In addition, since we only need to obtain the temperature curve of zero sensor output, the inclination is always about 90°; that is, θinc ≈ 90°. Here we assume that θinc = 90∘, and we experimentally found that this assumption has no obvious influence on the whole compensation procedure. ω is set to 1 in our model, and it is reasonable to consider it as a complete sine period when the sensor turns over from roll angle 0∘ to 360°. The experimental data in Figure 3 could also prove that. Therefore, (3) could be simplified as



Equation 4.



Therefore, as long as two points (t1, y1) and (t2, y2) are obtained, then k = Δy/(t2 − t1) and b=y1 − (Δy/(t2 − t1))·t1. Substitute k, b, and (4) into (2); then temperature compensation function of zero sensor output could be represented by



Equation 5.

3.Summary
Ericco provides high-precision Q-Flex accelerometer, such as the ER-QA-01A3, with a bias stability of 10μg, scale factor repeatability of 10ppm, and a weight of 80g, which can be widely used in aircraft carrier microgravity measurement systems, inertial navigation systems, and static angle measurement systems.