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Micromachined thermal gas inertial sensors based on heat convection are novel devices that compared with conventional micromachined inertial sensors offer the advantages of simple structures, easy fabrication, high shock resistance and good reliability by virtue of using a gaseous medium instead of a mechanical proof mass as key moving and sensing elements. This paper presents an analytical modeling for a micromachined thermal gas gyroscope integrated with signal conditioning. A simplified spring-damping model is utilized to characterize the behavior of the sensor. The model relies on the use of the fluid mechanics and heat transfer fundamentals and is validated using experimental data obtained from a test-device and simulation. Furthermore, the nonideal issues of the sensor are addressed from both the theoretical and experimental points of view. The nonlinear behavior demonstrated in experimental measurements is analyzed based on the model. It is concluded that the sources of nonlinearity are mainly attributable to the variable stiffness of the sensor system and the structural asymmetry due to nonideal fabrication.

The development of micromachined inertial sensors has been widely addressed for many years. Typical inertial sensors are based on the movement of a seismic proof mass caused by an inertial quantity. These sensors utilize different sensing principles: capacitive, piezoresistive and piezoelectric measurements [

A mechanism analysis along with mathematical modeling is an essential part of the required work in the sensor design and sensor optimization processes, especially for an inertial device. An analytical model often helps to understand the behavior of a device and resolve any concurrent problems. For example, an inertial sensor generally has nonlinear problems that usually lower the sensitivity and narrow the working range of the device. In order to get rid of these problems, many researchers have taken great efforts to investigate the nonlinear mechanisms and identify the nonideal sources by modeling [

In this paper, theoretical and experimental studies on characterization of a micromachined thermal gas gyroscope are presented. For the first time, a characterization of the sensor incorporating its signal conditioning using a simplified model of a spring-damping system is proposed and experimental verification is demonstrated. The modeling approach relies on the fundamentals of fluid mechanics and heat transfer, in association with empirical techniques. The proposed compact model is effective to handle the complexity of the device optimization. The experimental data are provided from both of model-based simulations and physical measurements using fabricated prototypes. The nonlinear characteristics of the sensor are analyzed based on the model and the nonideal sources are summarized.

A conceptual design of a micromachined gas gyroscope is shown in

The working principle of the device is based on the phenomenon of natural convection. A convectional flow is generated by heating the suspended central heater. For instance, when the central heater heats up and acceleration is applied on the direction of the Z-axis, a gas flow is generated in the region of the hermetic chamber and depicted in _{z}_{c}

The entire working process of the sensor consists of multi-physics interactions: electrical-thermal conversion, heat transfer, flow convective movement, and fluid-electrical conversion. A block diagram of the system model, including heating source, gas conduction, gas convection, and sensing, is shown in

Firstly, we consider the heating source. The electric power supplied to the heating resistor is dissipated by heat transfer toward the ambient fluidic medium and also toward the substrate (heating resistor), and which leads to a temperature difference between the heater and ambience. According to the Energy Principle [_{h}_{a}_{h}_{0} is a constant coefficient depending on the geometrical parameters of the heater. According to linear perturbation theory, the heat transfer coefficient _{1} = _{0}, _{1} = 1_{0}.

Then, we analyze the process of gas conduction. The gas conduction is the heat conduction. The temperature difference between the heater and external ambience leads to a heat transfer in the gaseous medium in the chamber. According to the heat transfer principle [_{2} are the gas density, specific heat, and thermal conductivity, respectively. The vector operator ∇ is defined as
_{h}_{2} = −_{0}(^{2}/(_{2}_{0}(_{0}(

In the process of gas convection, the gradient pressure is generated by the gradient temperature in terms of the state equation
_{w}_{3} = −_{0}(^{2}_{0}(_{3} = −_{2}∇_{0}(_{0}(_{0}(

Following the gas momentum equation and Archimedes’s law, an applied acceleration results in a buoyancy force and deforms the temperature profile [_{z}_{c}_{z}_{r}_{D}_{c}_{4} and _{4} are constant coefficients depending on thermal and fluidic properties of gas.

The thermistors convert the thermal signals (local temperatures) into the resistance signals of the resistors. Due to thermal inertia of the thermistors, another first-order transfer function should be considered since thermistors have to be in equilibrium with the local temperature of the gas to convert temperature variations into electrical resistance variations. The first-order transfer function represents the signal transfer from the local temperature difference Δ_{D}_{d}_{5} represents the time constant of the thermal inertia of the thermistors.

Using a Wheatstone bridge circuit, the temperature difference on the detecting thermistors is proportionally converted into a voltage difference

Combining the _{h}_{h}_{1}_{3}_{4}/_{1}_{2} + _{1}_{3} + _{1}_{4} + _{1}_{5} + _{2}_{3} + _{2}_{4} + _{2}_{5} + _{3}_{4} + _{3}_{5} + _{4}_{5}.

In practice, the coefficients

Extract the amplitude and phase of the output response as:

As explained in Section 2, the sensor output signal is detected using a synchronous demodulation technique, which can greatly eliminates disturbances and reduces noise level so as to enhance the accuracy and sensitivity of the sensor. The heating power is modulated at the frequency of _{0}Ω_{z}_{z}

The preceding analyses are based on the assumption of ideal gas and ideal device-structure. However, the practical conditions are complex and in general not ideal. The considered nonideal aspects affecting the device are mainly as follows: inaccurate phase–shift, asymmetrical structure due to unsatisfied fabrication, nonlinear dependence between temperature differences across detectors and Coriolis acceleration.

The first nonideal factor is an improper phase shift in the local reference oscillator due to improper electronic circuits, which will reduce the scale factor of the sensor (

The second nonideal factor affecting the sensor output is structural asymmetry in the chamber, heater, and detectors (

The third nonideal source comes from nonlinear dependence of temperature difference across detectors on Coriolis acceleration. A similar nonlinear phenomenon was found in a thermal accelerometer based on heat conduction [_{z}_{z}

To validate the effectiveness of the model established above, we conducted experiments using a device prototype shown in

The prepared sensor (device A) was mounted on a controlled rotary table. The Z-axis of the sensor was aligned vertically so that the Earth’s gravity acceleration was applied on the Z-axis of the sensor. The angular rate ranging from −600 deg/sec up to +600 deg/sec was applied around the Z-axis of the sensor. The output voltages of the sensor under a modulation/demodulation frequency of 8 Hz (

To investigate matters of nonlinearity, we further conducted a dual-phase demodulation measurement on the device and used the established model to simulate the output of the sensors. In the dual-phase measurement, two orthogonal reference signals with the same frequency and a phase difference of 90° were used to multiply the detected signal to obtain two orthogonal components of the output vector: _{0} ·[_{z}_{0} · [_{z}^{3} to 1.81 × 10^{3} with the increase of the magnitude of angular rate, and an asymmetrical coefficient

To further test the nonlinearity dependence on the stiffness and structural asymmetry, we used another device (device B) with a serious nonlinear feature to conduct experiments. The experimental setup for the device B was same as that for the device A. The dual-phase measurements were used once again in this experiment.

The model parameters were identified by fitting the measurement data. For device B, the equivalent damping coefficient ^{3} up to 6.08 × 10^{3} with the increase of angular rate, and an asymmetrical coefficient

The structural asymmetry also brings on a zero offset existing in the sensor output as shown in _{0} = _{0}_{0}_{0}

From the preceding measurements and analyses, it is seen that the model established in the paper can characterize well the performance of the sensor, and is feasible to be used for the optimal design and device improvement. It is also seen that the structural symmetry in the device is crucial for the linearity. The fabrication needs to be improved to amend structural asymmetry for eliminating the nonlinearity of the sensor. Besides the linearity of the sensors, we also tested noise limited resolution of the angular rate for the sensors. We found the noise densities of the sensors were around

A mathematical model (simplified as a spring-damping system) is established for a micromachined thermal gas gyroscope based on convection heat transfer to characterize multi-physics interaction processes: electrical-thermal conversion, convection heat transfer, flow convective activity and fluid-electrical conversion. A signal detection process using a modulation/demodulation technique is considered in the modeling, where the heating power is modulated at a given frequency and the angular rate is extracted by demodulation and a low-pass filter. The established model is validated by comparing the simulated results with the real measured data from dual-phase measurements. The theoretical and experimental studies reveal that the nonideal effects in the device are mainly attributable to the structural asymmetry and the variable stiffness of the system; the linearity of the sensor can be improved via amending the structural asymmetry in fabrication.

Conceptual design of a thermal gas gyroscope.

The convection field in region of hermetic chamber driven by heating the central heater under an acceleration along Z-axis.

Block diagram of signal transfers in the thermal gas gyroscope.

Block diagram of the sensor model.

Fabricated sensor prototype without packaging.

Output voltage of the sensor

Tow orthogonal components of the output vector

Tow orthogonal components of the output vector

Simulation results of two orthogonal components of the output vector

Two orthogonal components of the zero output