6.3: Acceleration Sensors

Learning Objectives

After completing this section, you will be able to:

1. Describe the working principle of a piezoelectric accelerometer.
2. Explain the working principle of a capacitance-based accelerometer.
3. Introduce the MEMS-based (micro-electromechanical systems) accelerometer.
4. Conduct coordinate transformation of the two- and three-dimensional accelerometer signals.

This section discusses how to measure the acceleration of an objector using an acceleration sensor called an accelerometer. Measuring an objector’s acceleration is very important. It is a critical physical variable that can be used, for example, for feedback control, calculating other variables (such as a pendulum angle), and measuring the movement of a cell phone or a remote controller. Note that the accelerometer is based upon the measured inertia force \(m\ddot{x}\), where \(m\) is the sensing mass and \(\ddot{x}\) is the acceleration of the mass. The main difference in accelerometer working principles is the way the inertia force was measured. The accelerometer (acceleration sensor) introduced in Section 2.4, an acceleration sensor, is features in Figure 6.12. It uses a piezoelectric force measurement device to obtain the system acceleration.

Section 2.4 discussed the transfer function from the sensing element acceleration \(\ddot{x}\) to output voltage \(V_p(s)\). The Laplace transform for the acceleration \(\ddot{x}\) is \(s^2X(s)\), where \(X(s)\) is the Laplace transform of \(x\). That transfer function was shown in equation \ref{2.134} and is also shown below.

\[G_{ap}\left(s\right)=\dfrac{V_p(s)}{s^2X(s)}=\dfrac{V_{p-norm}ω_{min}s}{s+ω_{min}}\cdot \dfrac{ω^2_{max}\left(ms^2+bs+k\right)/m}{s^2+2ζω^2_{max}s+ω^2_{max}}\cdot k \label{6.17}\]

where the parameter definition of the above transfer function can be found in Section 2.4 and reference [23]. The frequency response of the transfer function in equation \ref{6.17} was plotted earlier in Figure 2.38. Note that the DC gain of this transfer function is zero because there is a differentiation component \(s\) in the numerator of the transfer function \ref{6.17}. The DC gain can be obtained by setting \(s=0\), which leads to the following.

\[G_{ap}\left(0\right)=0 \label{6.18}\]

That is, for constant acceleration, the output of the piezoelectric acceleration sensor is zero, which is due to the characteristics of the piezoelectric material. Under constant pressure, there is no new energy supplied to the piezoelectric material. As a result, there is no electric piezoelectric energy (signal) output. Therefore, the piezoelectric accelerometer is good for measuring transient acceleration, but not good for measuring steady-state (constant) acceleration. This distinction is very important for selecting a proper accelerometer for a specific application.

Next, we will discuss sensing technology for capacitance-based acceleration. This sensing technology is also based upon the principle of inertia force. By this principle, a displacement is generated against a spring due to the acceleration of the measuring object. This displacement results in movement of a plate between the two electrodes. Figure 6.13 shows the architecture of such a sensor.

Assume that without the acceleration (at rest), the displacements \(d_1=d_2=d\) and the moving mass displacement \(x=0\). When \(x\neq 0\), we have the following.

\[d_1=d-x, d_2=d+x \label{6.19}\]

Based upon equation \ref{6.19}, we can summarize the relationships of displacements (\(d_1\) and \(d_2\)) and the capacitance values (\(C_1\) and \(C_2\)) in Table 6.2. Note that capacitance values \(C_1\) and \(C_2\) are inversely proportional to the associated distances \(d_1\) and \(d_2\), respectively. That is, when the distance increases, the capacitance reduces.

Note that capacitance is the capacity of a device to store an electric charge (energy). It can be measured by the charge in response to a difference in electric potential, expressed as the ratio of those quantities. There are essentially two closely related capacitances: self-capacitance and mutual capacitance [24].\({}^{\mathrm{\ }}\) An object capable of being electrically charged exhibits self-capacitance, for which the electric potential is measured between the object and ground. Mutual capacitance is measured between two components (or plates), which is particularly important for these capacitors in the electronic circuits.

Table 6.2: Displacements (\(d_1\) and \(d_2\)) and the associated capacitance values (\(C_1\) and \(C_2\)).

\(x>0\)	\(d_1<d_2\)	\(C_1>C_2\)
0\)" style="text-align:left;" class="lt-eng-137571" data-id="lt-eng-137571">\(x=0\)	\(d_1=d_2\)	C_2\)" style="text-align:left;" class="lt-eng-137571" data-id="lt-eng-137571">\(C_1=C_2\)
0\)" style="text-align:left;" class="lt-eng-137571" data-id="lt-eng-137571">\(x<0\)	\(d_1>d_2\)	C_2\)" style="text-align:left;" class="lt-eng-137571" data-id="lt-eng-137571">\(C_1<C_2\)

Consider a capacitor with two parallel plates. The capacitance \(C\) (dependent on its geometry) is given by the formula below.

\[C=\dfrac{ρA}{d} \label{6.20}\]

where \(A\) is the area of each plate; \(d\) is the distance between the plates; and \(ρ\) is the permittivity of the material between the plates of the parallel capacitor, where the permittivity of free space, \({ρ=ρ}_0\), is a physical constant approximately equal to \(ρ_0=8.854\times {10}^{-12}\) Farads per meter (\(F/m\)). Equation \ref{6.20} indicates that the capacitance is inversely proportional to the distance between the two plates.

Example 6.5

Calculate the capacitance of a two-plate capacitor with distance \(d=0.2\ mm\) and plate area of \(2\ {mm}^2\), with vacuum between the two plates. The capacitance can be calculated using equation \ref{6.20} as below with the approximated permittivity of free space for air permittivity.

\[ \ C=\dfrac{ρA}{d}=\dfrac{8.854\times {10}^{-12}\ \left(F/m\right)\ \times 2\ \left({mm}^2\right)}{0.2\ mm}=8.854\times {10}^{-14}\ F=0.08854\ pF \label{6.21}\]

For the case indicated in Figure 6.13, the capacitance difference \(C_2-C_1\) can be calculated below based on equation \ref{6.19}.

\[C_2-C_1=\dfrac{ρA(d_1-d_2)}{d_2d_1}=\dfrac{-2ρAx}{(d+x)(d-x)}=\dfrac{-2ρAx}{d^2-x^2} \label{6.22}\]

The relative permittivity of air in a two-plate capacitor is \(1.0006\) at standard temperature and pressure. However, it can be rounded off to \(1.0\) for practical purposes, or \(ρ=ρ_0\). This is because air is a gas with very few molecules between the capacitor plates. Therefore, it acts like the capacitor plates in a vacuum.

We can now derive the transfer function for a capacitance-based accelerometer. Figure 6.14 shows an equivalent electronic circuit for acceleration measurement, where \(C_1\) and \(C_2\) are the two capacitors defined in Figure 6.13. Two operational amplifiers are used. One has negative feedback to track the combined inputs \(v_s\) and \(-v_s\). The other is a following circuit to isolate the sensing circuit from the low-pass filter circuit as an anti-aliasing filter.

Let’s next derive the transfer function from \(V_s(s)\) to \(V_a(s)\), where \(V_s(s)\) and \(V_a(s)\) are the Laplace transforms of \(v_s(t)\) and \(v_a(t)\) defined in Figure 6.14, respectively. Using equation \ref{2.77}, we can calculate the transfer function from \(+V_s(s)\) to \(V_a(s)\) as below.

\[\dfrac{V_a(s)}{V_s(s)}=-\dfrac{Z_o\left(s\right)}{Z_i\left(s\right)}=-\dfrac{\dfrac{R_F/C_Fs}{R_F+1/C_Fs}}{1/C_1s}=-\dfrac{R_FC_1s~}{R_FC_Fs+1} \label{6.23}\]

In equation \ref{6.23}, \(Z_o\left(s\right)\) is a parallel connection of resistor \(R_F\) and impedance \(1/C_Fs\), and \(Z_i\left(s\right)\) is impedance \(1/C_1s\). The transfer function from \(-V_s(s)\) to \(V_a(s)\) can be found in a similar way as below.

\[\dfrac{V_a(s)}{-V_s(s)}=-\dfrac{Z_o\left(s\right)}{Z_i\left(s\right)}=-\dfrac{\dfrac{R_F/C_Fs}{R_F+1/C_Fs}}{1/C_2s}=-\dfrac{R_FC_2s~}{R_FC_Fs+1} \label{6.24}\]

In this case, \(Z_o\left(s\right)\) is the same as before, but \(Z_i\left(s\right)\) is the impedance \(1/C_2s\) (different in this case). To obtain the complete transfer function, equations \ref{6.23} and \ref{6.24} can be consolidated. The resulting equation can be found below; it is a linear circuit with two different inputs \(V_s(s)\) and \(-V_s(s)\) to the operational amplifier.

\[\dfrac{V_a(s)}{V_s(s)}=\dfrac{R_Fs~}{R_FC_Fs+1}\left(C_2-C_1\right) \label{6.25}\]

Power supplied to the capacitance-based accelerometer is constant (say \(v_s\)) with its transfer function below.

\[V_s(s)=\dfrac{v_s~}{s} \label{6.26}\]

Then, with measured \(V_a(s)\), we can calculate gain \(C_2-C_1\). Equation \ref{6.22} can be used to solve for displacement \(x\). Using the following relationship,

\[m\ddot{x}=-xk \label{6.27}\]

acceleration \(\ddot{x}\) can be calculated with \(k\) to be the spring stiffness defined in Figure 6.13, and \(m\) is the value of moving mass.

Now we will address how an accelerometer is realized in the form of a MEMS. Figure 6.15 shows an example of MEMS implementation of a one-directional accelerometer.

From Figure 6.15, we can see that a one-dimensional accelerometer can be implemented in a microchip with a moving part attached with electrode fins as well as two fixed parts attached with electrodes. The top and bottom bars are attached to the moving mass. They act as two springs similar to these in the architecture plot of Figure 6.13. Both ends of these two bars are anchored to the fixed base. The same is true for two sensing elements attached with electrode fins on the left and right sides of plot in Figure 6.13. Figure 6.15 also provides an equivalent circuit to mark both capacitors \(C_1\) and \(C_2\) defined in Figure 6.13. Recall that in Figure 6.13, when the mass moves up, capacitance \(C_1\) increases with reduced \(d_1,\) while at the same time, capacitance \(C_2\) decreases with increased \(d_2\).

For the sensing system shown in Figure 6.15, when the mass moves up, capacitance \(C_1\) increases with reduced distance between the two electrodes on the left. At the same time, capacitance \(C_2\) decreases with increased distance between the two electrodes on the right. In most of the MEMS accelerometers, three-dimensional accelerometers are implemented in a single chip (see Figure 6.16). In this case, the one-dimensional accelerometer shown in Figure 6.15 is replicated three times in three orthogonal directions. In addition, the electronic circuit shown in Figure 6.14 is also integrated into the same MEMS ship. Therefore, a MEMS ship itself is a mechatronic subsystem.

The installed sensor directions (body-coordinate: \(x,\ y,\ z\) shown in Figure 6.16) are often not in line with the physical coordinates. In this case, coordinate transformation is needed. First, consider the two-dimensional case (\(x\)-\(y\) plane) with an acceleration vector \(\tilde{a}\) shown in Figure 6.17.

The sensor measures (\(\tilde{a}_x,\tilde{a}_y\)) in the sensor coordinate, and (\( \tilde{a}_{x_p},\tilde{a}_{y_p}\)) in the physical coordinate (the vehicle coordinate in Figure 6.17) can be calculated by the following equation.

\[\left[ \begin{array}{c} \tilde{a}_{x_p} \\ \tilde{a}_{y_p} \end{array} \right]=\left[ \begin{array}{cc} {\cos θ } & -{\sin θ } \\ {\sin θ } & {\cos θ } \end{array} \right]\left[ \begin{array}{c} \tilde{a}_x \\ \tilde{a}_y \end{array} \right], \label{6.28}\]

where

\[\tilde{a}=\sqrt{\tilde{a}^2_x+\tilde{a}^2_y}=\sqrt{\tilde{a}^2_{x_p}+\tilde{a}^2_{y_p}} \label{6.29}\]

Note that \(θ\) is positive if the sensor coordinate rotates anticlockwise relative to the physical coordinate and can be either calibrated or obtained through a machine learning process.

For the three-dimensional case, the coordinate transformation can be divided into three sequential rotations: a) around \(z_p\) axis (\(θ\)), b) \(y_z\) axis (\(\phi\)), and c) \(x_y\) axis (\(ψ\)). Figure 6.18 shows these three rotations.

As a result, the coordinate transformation can be factored into three stages below.

\[{\left[ \begin{array}{c} \tilde{a}_{x_p} \\ \tilde{a}_{y_p} \\ \tilde{a}_{z_p} \end{array} \right]=R}_{yz}R_{zx}R_{xy}\left[ \begin{array}{c} \tilde{a}_{x_s} \\ \tilde{a}_{y_s} \\ \tilde{a}_{z_s} \end{array} \right] \label{6.30}\]

with rotation in \(z_p\) defined below

\[\left[ \begin{array}{c} \tilde{a}_{x_z} \\ \tilde{a}_{y_z} \\ \tilde{a}_{z_z} \end{array} \right]=\left[ \begin{array}{ccc} {cos θ\ } & -{sin θ\ } & 0 \\ {sin θ\ } & {cos θ\ } & 0 \\ 0 & 0 & 1 \end{array} \right]\left[ \begin{array}{c} \tilde{a}_{x_p} \\ \tilde{a}_{y_p} \\ \tilde{a}_{z_p} \end{array} \right]=R_{xy}\left[ \begin{array}{c} \tilde{a}_{x_p} \\ \tilde{a}_{y_p} \\ \tilde{a}_{z_p} \end{array} \right],\label{6.31}\]

rotation in \(y_z\) defined below

\[\left[ \begin{array}{c} \tilde{a}_{x_y} \\ \tilde{a}_{y_y} \\ \tilde{a}_{z_y} \end{array} \right]=\left[ \begin{array}{ccc} {\cos \phi \ } & 0 & {\sin \phi \ } \\ 0 & 1 & 0 \\ {-\sin \phi \ } & 0 & {\cos \phi \ } \end{array} \right]\left[ \begin{array}{c} \tilde{a}_{x_z} \\ \tilde{a}_{y_z} \\ \tilde{a}_{z_z} \end{array} \right]=R_{zx}\left[ \begin{array}{c} \tilde{a}_{x_z} \\ \tilde{a}_{y_z} \\ \tilde{a}_{z_z} \end{array} \right],\label{6.32}\]

and rotation in \(y_z\) defined below

\[\left[ \begin{array}{c} \tilde{a}_{x_s} \\ \tilde{a}_{y_s} \\ \tilde{a}_{z_s} \end{array} \right]=\left[ \begin{array}{ccc} 1 & 0 & 0 \\ 0 & {\cos ψ\ } & -{\sin ψ\ } \\ 0 & {\sin ψ\ } & {\cos ψ\ } \end{array} \right]\left[ \begin{array}{c} \tilde{a}_{x_y} \\ \tilde{a}_{y_y} \\ \tilde{a}_{z_y} \end{array} \right]=R_{yz}\left[ \begin{array}{c} \tilde{a}_{x_y} \\ \tilde{a}_{y_y} \\ \tilde{a}_{z_y} \end{array} \right]\label{6.33}\]

Example 6.6

Find the two-dimensional coordinate transformation matrix defined in equation \ref{6.27} if the rotational angle \(θ=30^\circ\). The matrix can be calculated as follows.

\[T_p=\left[ \begin{array}{cc} {\cos 30^\circ  } & -{\sin 30^\circ  } \\ {\sin 30^\circ \ } & {\cos 30^\circ \ } \end{array} \right]=\left[ \begin{array}{cc} 0.866 & -0.5 \\ 0.5 & 0.866 \end{array} \right] \label{6.34}\]

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