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Micro-mechanical systems (MEMS) have become common. They are used for acceleration measurement in a multitude of designs, from in-vehicle systems to fitness trackers. Figure 1 shows the operation principle of an accelerometer. The basis of such a sensor is a movable proof mass m, fixed on suspensions with a coefficient of elasticity k. The capacitors with movable plates are connected to the mass and the system body in parallel to these suspensions.
The displacement of the mass, according to Hooke’s law, is proportional to its acceleration. In the case of small displacements of a mass, a linear dependence is observed between displacement and the voltage at the bridge circuit, combined from two capacitors with movable plates. Consequently, the mechanical model of an accelerometer follows the rules of forced oscillations and described by the formula:
X ̈+r/m X ̇+k/m X=F/m cos(ωt)
The natural oscillation frequency (or resonant frequency) and Q-factor are determined by the formula:
ω_0=√(k/m), (2), Q=(m∙ω_0)/k
Thus, the transfer function in the Laplace transform will look as follows:
H_M (s)=〖ω_0〗^2/(s^2+(ω_0/Q)∙s+〖ω_0〗^2 )
In some accelerometers with analogue output, an additional low pass filter is built in to reduce the gain at the resonant frequency. For this purpose, the original transfer function is multiplied by a low pass filter term with a cutoff frequency ωC:
H(s)=H_M (s)∙ω_C/(s+ω_C )
It is common for a mechanical system analysis to use an LRC-filter analogy. In this case, the Laplace transform formula, dependencies between natural frequency ω0 and Q-factor from on side and LRC-filter parameters from another can be shown as:
H_M (s)=1/(〖L∙C∙s〗^2+R∙C∙s+1), L∙C=1/〖ω_0〗^2 , R∙C=1/(Q∙ω_0 )
MEMS accelerometer modelling
Modelling a MEMS accelerometer in LTspice is possible using the Laplace transform functional of the simulation tool. To achieve this using a voltage-controlled source (Spice prefix B), add the Laplace function to the Value2 string (Figure 2).
Figure 2: Inserting Laplace formula into Value2 string of a voltage-controlled voltage source
Figure 3: The ADXL365 accelerometer model with Laplace X and Y axes parameters
Figure 4: ADXL356XY model’s frequency response, showing good agreement with accelerometer’s datasheet
The frequency response of such a circuit is in good agreement with the ADXL365’s supplier datasheet. The result of the simulation in logarithmic scale is shown in Figure 4.
Laplace function parameters can be calculated for different accelerometers, a summary of some sensors from Analog Devices are listed in Table.
Vibrational sensor signal chain modelling
The vibrational sensor model can be used for a comprehensive assessment of a combined signal chain. This signal chain includes an accelerometer and an analogue filter used at the sensor output to bridge it with the rest of the system (normally with an adc). It is common to use a low noise operational amplifier in such analogue filter circuits. In this case this operational amplifier is normally used as a low pass filter. This implementation can help to reduce a resonance peak value and expand the range of vibration spectral measurements (within a 3dB range). Figure 6 shows an example of the accelerometer model with a low pass filter based upon an operation amplifier. The modelling result confirms that the measurement range is increased by additional value within 3dB (Figure 5).
Figure 5: Using a low pass filter allows an increase in measurement range of the combined sensor system
Figure 6: ADXL1002 accelerometer model with a buffer used to model a combined signal chain
Figure 7: Modelling noise level
This technique also allows us to estimate the noise performance of the system. For a rough approximation, the MEMS accelerometer has noise with a uniform spectral density. The ADXL1002 spectral noise floor is estimated at 25-ug/√Hz. At a sensor supply voltage of 5V, spectral density will be expressed in voltage noise at the level of 1.25 uV/√Hz. To model such uniform noise an engineer can use a resistor with a value R=9002-Ω and a voltage-controlled voltage source with a gain ratio of 1,000 (Figure 7).
The calculated noise spectrum shows that the accelerometer noise will dominate at low frequencies, and the amplification circuit at high frequencies (Figure 8).
Figure 8: Spectral noise at the accelerometer output
Using Laplace transform functionality enables calculations in the simulator for evaluation and optimisation of the analogue signal chain circuit, including its noise characteristics. This calculation yields an optimal choice of the amplifying circuit, which allows an engineer to expand the measurement range of the MEMS accelerometer and get closer to the characteristics of a more expensive piezo-sensor, broadening its appeal for cost-sensitive applications.
