EE3901/EE5901 Sensor Technologies
Week 12 Notes
Piezoelectricity and accelerometers

College of Science and Engineering, James Cook University

Last updated: 16 May 2023

Piezoelectricity is a direct relationship between mechanical stress and electric polarisation. It can be used to build both sensors and actuators. This week we will study the piezoelectric effect and then apply it to the problem of sensing. For example, an important industrial application of piezoelectricity is to make high performance accelerometers.

The piezoelectric effect
More about stress
A mathematical description of piezoelectricity
Interface circuit considerations
Accelerometers
References

The piezoelectric effect

Piezoelectric materials exhibit a reversible relationship between mechanical stress and electric polarisation. When mechanical stress is applied, the material becomes polarised. Similarly, when an electric field is applied, the material deforms.

A “toy model” to understand piezoelectricity is as follows. Figure 1 (a) shows a simplified crystal structure where atoms are arranged in a hexagon. Suppose that each atom in the crystal has partial charge (e.g. due to differing levels of electronegativity causing unequal sharing of electrons). In the absence of any stress, the structure is symmetric and there is no electric polarisation.

Figure 1: An intuitive understanding of how piezoelectricity can occur. (a) In the absence of any mechanical stress, the “toy model” crystal is symmetric and there is no net electric polarisation. (b) Deforming the material breaks the symmetry, here causing the left hand side to be more positive and the right hand side to be more negative. Zoom:

When mechanical stress is applied, the material deforms, as shown in Figure 1 (b). The symmetry is disrupted, and there is a net polarisation in a particular direction. Notice that this is not necessarily the same direction as the applied stress.

A well known piezoelectric material is quartz, which is a naturally occurring mineral that is abundant in the earth. You may have come across quartz crystal resonators used to generate clock signals for digital electronics. Quartz oscillators are also common in wristwatches.

Another common piezoelectric material is the synthetically produced ceramic PZT (lead zirconate titanate). There are many other materials that have been discovered to have piezoelectric properties.

More about stress

Previously when we analysed strain gauges we used the symbol $\sigma$ for stress and $\epsilon$ for strain. However, $\epsilon$ is also used for permittivity. Therefore, the piezoelectric literature often uses $S$ for strain and $T$ for stress. We will follow that notation here.

To write a mathematical description of piezoelectricity, we first need to understand stress and strain in more detail. We will first consider a simple two dimensional geometry and define normal and shear stress. Then we will extend that to three dimensions.

Normal stress in 2D

Consider the differential volume element shown in Figure 2. We label the faces according to the direction of the normal vector, i.e. face x is the face where the normal vector points in the $x$ direction. Figure 2 shows a stress $T_{xx}$ . The subscripts mean that the stress acts on face x in direction $x$ .

All our analysis of stress and strain will be conducted under the assumption of static equilibrium. Therefore, all forces must sum to zero. Figure 2 shows two equal stresses pointing in opposite directions. It is clear that this configuration is statically stable because the forces cancel out.

This geometry is called normal stress. It occurs when the direction of the stress lies along the normal vector.

Figure 2: A volume element experiencing normal stress. Zoom:

Shear stress in 2D

Another type of stress is called shear stress. It occurs when the force occurs along some direction other than the normal to the face. An example is shown in Figure 3.

In Figure 3 (a), a stress $T_{yx}$ acts on face y in direction $x$ . It should be clear that this arrangement is not statically stable. There is a torque generated by the unsymmetrical arrangement of the forces. Hence, such a configuration would begin to rotate. To maintain equilibrium, an additional pair of forces must be applied.

Figure 3 (b) shows a statically stable arrangement of shear stress. The condition for equilibrium is $T_{xy} = T_{yx}$ .

Figure 3: An example of shear stress. (a) An arrangement of stresses that violates the assumption of static equilibrium. (b) Static equilibrium requires that forces and torques both sum to zero. Zoom:

The general case

Let’s now extend these ideas to 3D. In piezoelectric literature, it is common to label the axes as 1, 2, and 3. Moments around each axis are numbered 4, 5, and 6. The coordinate system is shown in Figure 4 (a). Faces are named after their normal vectors, as per Figure 4 (b). The sign convention is that positive stresses act in the direction of the axis on the face where the normal is positive. If the normal is negative, then positive stresses act in the opposite direction. Hence, we can see from Figure 4 (b) that positive stresses act in tension and negative stresses act in compression.

A stress $T_{ij}$ is the stress acting on face $i$ in direction $j$ . The condition of static equilibrium requires that $T_{ij} = T_{ji}$ . Therefore, despite there being 9 possible directions of stress, the constraints leave only 6 degrees of freedom. This leads to the definition of the Voigt notation for stresses. In Voigt notation, we label the stresses by a single number. The normal stresses are given by

\begin{align*} T_{1} & = T_{11}\\ T_{2} & = T_{22}\\ T_{3} & = T_{33}. \end{align*}

Similarly, we name the shear stresses based on the axis of their moment:

\begin{align*} T_{4} & = T_{23} = T_{32}\\ T_{5} & = T_{13} = T_{31}\\ T_{6} & = T_{12} = T_{21}. \end{align*}

Figure 4: (a) The coordinate system used to describe stress, strain, and piezoelectricity. (b) Faces are named after their normal vector. (c) The notation with two numbers refers to a face and a direction, whereas notation with one number refers to the axes defined in part (a). (d) Normal stresses are defined as indicated, with positive stress being tension and negative stress being compression. (e) Shear stresses consist of two equal, opposing moments around the indicated axes. Zoom:

A mathematical description of piezoelectricity

The piezoelectric strength of a material is measured by coefficients $d_{ij}$ in units of C/N. In the general case $\left[d\right]$ is a $3\times6$ matrix, although in practical materials many of the coefficients are zero due to the underlying crystal symmetries.

There are two types of piezoelectric effect. These are called the direct effect (used for sensors) and the converse effect (used for actuators).

The direct effect

The direct effect is the appearance of electric polarisation due to an applied mechanical force,

\begin{equation} \boldsymbol{D} =\left[d\right]\boldsymbol{T}+\left[\epsilon\right]\boldsymbol{E}, \end{equation}

where $\boldsymbol{D}$ is the electric displacement (units of $\text{C}/\text{m}^{2}$ ), $d$ is the piezoelectric coefficient (units of $\text{C}/\text{N}$ ), $\boldsymbol{T}$ is stress (units of $\text{N}/\text{m}^2$ ), $\epsilon$ is the permittivity (which in the general case is different along different axes, hence why it is represented as a matrix), and $\boldsymbol{E}$ is the electric field (units of $\text{V}/\text{m}$ ). Some authors write the permittivity as $\epsilon_{ij}^{T}$ where the superscript $T$ means “measured at constant stress”.

Writing out Eq. (1) in full,

\left[\begin{array}{c} D_{1}\\ D_{2}\\ D_{3} \end{array}\right]=\left[\begin{array}{cccccc} d_{11} & d_{12} & d_{13} & d_{14} & d_{15} & d_{16}\\ d_{21} & d_{22} & d_{23} & d_{24} & d_{25} & d_{26}\\ d_{31} & d_{32} & d_{33} & d_{34} & d_{35} & d_{36} \end{array}\right]\left[\begin{array}{c} T_{1}\\ T_{2}\\ T_{3}\\ T_{4}\\ T_{5}\\ T_{6} \end{array}\right]+\left[\begin{array}{ccc} \epsilon_{11} & 0 & 0\\ 0 & \epsilon_{22} & 0\\ 0 & 0 & \epsilon_{33} \end{array}\right]\left[\begin{array}{c} E_{1}\\ E_{2}\\ E_{3} \end{array}\right].

This equation allows us to calculate the electric response for a given amount of mechanical stress.

The converse effect

The converse effect is the appearance of mechanical strain due to an applied electric field,

\begin{equation} \boldsymbol{S}=\left[d\right]^{T}\boldsymbol{E}+\left[s\right]\boldsymbol{T}, \end{equation}

where $\boldsymbol{S}$ is the strain, $d$ is the piezoelectric coefficient matrix, $\boldsymbol{E}$ is the electric field, $s$ is the elastic compliance matrix (the reciprocal of the Young’s modulus), and $\boldsymbol{T}$ is the stress.

Writing out Eq. (2) in full,

\left[\begin{array}{c} S_{1}\\ S_{2}\\ S_{3}\\ S_{4}\\ S_{5}\\ S_{6} \end{array}\right]=\left[\begin{array}{ccc} d_{11} & d_{21} & d_{31}\\ d_{12} & d_{22} & d_{32}\\ d_{13} & d_{23} & d_{33}\\ d_{14} & d_{24} & d_{34}\\ d_{15} & d_{25} & d_{35}\\ d_{16} & d_{26} & d_{36} \end{array}\right]\left[\begin{array}{c} E_{1}\\ E_{2}\\ E_{3} \end{array}\right]+\left[\begin{array}{cccccc} s_{11} & \ldots\\ \ldots & s_{22} & \ldots\\ & \ldots & s_{33} & \ldots\\ & & \ldots & s_{44} & \ldots\\ & & & \ldots & s_{55} & \ldots\\ & & & & \ldots & s_{66} \end{array}\right]\left[\begin{array}{c} T_{1}\\ T_{2}\\ T_{3}\\ T_{4}\\ T_{5}\\ T_{6} \end{array}\right].

Notice that the piezoelectric constant matrix $\left[d\right]$ is transposed. Some authors write the elastic compliance elements as $s_{ij}^{E}$ to emphasise that they are measured at constant electric field.

Piezoelectric effect summary

The two effects can be summarised as

\begin{align*} \text{Direct effect (sensor equation): }\boldsymbol{D} & =\left[d\right]\boldsymbol{T}+\left[\epsilon\right]\boldsymbol{E}\\ \text{Converse effect (actuator equation): }\boldsymbol{S} & =\left[d\right]^{T}\boldsymbol{E}+\left[s\right]\boldsymbol{T}. \end{align*}

In practice many of the components of $\left[d\right]$ are zero. The specific arrangement of nonzero elements depends upon the material’s crystal structure. For example, many piezo ceramics have what is called tetragonal symmetry, which means there are only 5 nonzero components ( $d_{31}$ , $d_{32}$ , $d_{33}$ , $d_{15}$ and $d_{24}$ ). Furthermore, these are constrained so there are only 3 degrees of freedom: $d_{31}=d_{32}$ and $d_{15}=d_{24}$ . For that system, all other $d$ values are zero. For these ceramics, the direction of axis 3 is defined during manufacturing by applying an electric field at high temperature, which causes polarisation of internal dipoles inside the material. This process is called “poling”.

For sensing purposes, piezoelectric materials measure strain. If the device is under open circuit conditions (i.e. we simply measure the voltage across the device using a circuit with a very large input impedance), then the electric displacement is given by $\boldsymbol{D}=0$ . We can then use the sensor equation to calculate the electric field and hence the open circuit voltage. Recall that the electric field is measured in volts per meter. Assuming plane parallel geometry, $E_{i}=V/L_{i}$ where $L_{i}$ is the distance between the electrodes.

Example 10.1

A 1 cm $\times$ 1 cm $\times$ 1 mm slab of PZT (lead zirconate titanate) is resting on a rigid surface. A force of 1 N in compression is applied along the 3rd axis, as shown in Figure 5.

By convention for this class of material, it has been poled along axis 3. The following material properties are known:

\begin{align*} d_{33} & =370\ \text{pC/N}\\ d_{31}=d_{32} & =-110\ \text{pC/N}\\ d_{15}=d_{24} & =420\ \text{pC/N}\\ \epsilon_{33} & =1200\epsilon_{0}\\ \epsilon_{11}=\epsilon_{22} & =1250\epsilon_{0}. \end{align*}

Figure 5: A slab of PZT ceramic experiences a force as indicated. Zoom:

For voltage measurements, electrodes have been deposited on each face of the material (with a gap at the edges to avoid shorting them together). Voltage meters are connected as shown.

(a) Calculate the open circuit voltages $V_{1}$ , $V_{2}$ , and $V_{3}$ .

(b) How would the results change if the force was applied in tension instead of compression?

Solution

(a) First, calculate the applied strain,

T_{3}=\frac{-1\ \text{N}}{\left(0.01\ \text{m}\right)^{2}}=-10^{4}\ \text{N}/\text{m}^{2}.

The negative sign appears because the force is directed into the material (i.e. the material is in compression). All other strain values are 0. Furthermore, since the device is at open circuit, we have $\boldsymbol{D}=0$ . Writing the sensor equation for this system

\left[\begin{array}{c} D_{1}\\ D_{2}\\ D_{3} \end{array}\right]=\left[\begin{array}{c} 0\\ 0\\ 0 \end{array}\right]=\left[\begin{array}{cccccc} 0 & 0 & 0 & 0 & d_{15} & 0\\ 0 & 0 & 0 & d_{24} & 0 & 0\\ d_{31} & d_{32} & d_{33} & 0 & 0 & 0 \end{array}\right]\left[\begin{array}{c} 0\\ 0\\ T_{3}\\ 0\\ 0\\ 0 \end{array}\right]+\left[\begin{array}{ccc} \epsilon_{11} & 0 & 0\\ 0 & \epsilon_{22} & 0\\ 0 & 0 & \epsilon_{33} \end{array}\right]\left[\begin{array}{c} E_{1}\\ E_{2}\\ E_{3} \end{array}\right].

Reading off the first row,

\begin{align*} 0 & =0+\epsilon_{11}E_{1}\\ E_{1} & =0. \end{align*}

Therefore there is no electric field in the 1 direction, and hence $V_{1}=0$ .

Reading off the second row,

\begin{align*} 0 & =0+\epsilon_{22}E_{2}\\ E_{2} & =0. \end{align*}

Again we have $V_{2}=0$ .

Finally the third row gives

\begin{align*} 0 & =d_{33}T_{3}+\epsilon_{33}E_{3}\\ & =370\times10^{-12}\times\left(-10^{4}\right)+1200\times8.85\times10^{-12}\times E_{3}\\ E_{3} & =348.4\ \text{V/m}. \end{align*}

This corresponds to a voltage of $V_{3}=348.4\ \text{V/m}\times0.001\ \text{m}=0.3484\ \text{V.}$

(b) The voltage sign is defined relative to the direction of axis 3. If the force were applied in tension then the voltage would have the opposite sign.

Example 10.2

The same device from Example 10.1 is now used as an actuator. There is no mechanical resistance to expansion or contraction, i.e. the stress is zero. Calculate the strain and the resulting change in dimensions when a voltage $V_{3}=5\ \text{V}$ is applied to the device.

Solution

The applied electric field is $E_{1}=0$ , $E_{2}=0$ , and $E_{3}=5\ \text{V}/0.001\ \text{m}=5000\ \text{V/m}$ .

Using the actuator equation with $\boldsymbol{T}=0$ we have

\left[\begin{array}{c} S_{1}\\ S_{2}\\ S_{3}\\ S_{4}\\ S_{5}\\ S_{6} \end{array}\right]=\left[\begin{array}{ccc} 0 & 0 & d_{31}\\ 0 & 0 & d_{32}\\ 0 & 0 & d_{33}\\ 0 & d_{24} & 0\\ d_{15} & 0 & 0\\ 0 & 0 & 0 \end{array}\right]\left[\begin{array}{c} 0\\ 0\\ 5000 \end{array}\right].

Hence,

\left[\begin{array}{c} S_{1}\\ S_{2}\\ S_{3}\\ S_{4}\\ S_{5}\\ S_{6} \end{array}\right]=\left[\begin{array}{c} 5000d_{31}\\ 5000d_{32}\\ 5000d_{33}\\ 0\\ 0\\ 0 \end{array}\right]=\left[\begin{array}{c} -5.5\times10^{-7}\\ -5.5\times10^{-7}\\ 1.85\times10^{-6}\\ 0\\ 0\\ 0 \end{array}\right].

There is no shear strain in this material, only normal strain. Along axes 1 and 2 there is a length change of

\Delta L_{1}=L_{1}\times\left(-5.5\times10^{-7}\right)=0.01\times\left(-5.5\times10^{-7}\right)=-5.5\ \text{nm}.

Conversely along axis 3 there is a length change of

\Delta L_{3}=L_{3}\times1.85\times10^{-6}=0.001\times1.85\times10^{-6}=1.85\ \text{nm}.

Piezoelectric materials can be used to build nanometer scale actuators for applications like optics, microscopy, etc.

Interface circuit considerations

An equivalent circuit model for a piezoelectric sensor is shown in Figure 6.

Figure 6: An equivalent circuit model of a piezoelectric sensor. Zoom:

The piezoelectric effect creates a voltage $V_{i}$ . When $V_{i}$ is time-varying then it will displace charge onto the capacitive electrodes. This charge will slowly drain via $R_{s}$ , which represents the Ohmic conductivity of the piezoelectric material itself. This model clearly shows that piezo sensors only respond to time-varying signals. They have no DC response.

The lowest usable frequency can be determined from the circuit model. Notice that $R_{s}$ and $C$ form a highpass filter. Given the material resistivity and permittivity and the sensor geometry, you can calculate $C$ and $R_{s}$ and hence find the -3 dB cut-off frequency

f_{min}=\frac{1}{2\pi R_{s}C}.

Piezoelectric materials also have an upper limit of frequency because of mechanical resonance effects. The piezoelectric material behaves like a very stiff spring: it can store mechanical energy in the form of stress. It can also store electrical energy due to the electrode capacitance. The electrical and mechanical variables are coupled together. Two coupled energy storage elements is the recipe for resonance, i.e. there is some frequency at which input energy from $V_{i}$ will accumulate in the system, causing the output to grow larger and larger until non-linearities cause enough damping to clamp the amplitude of oscillation.

A typical frequency response for a piezoelectric sensor is shown in Figure 7.

Figure 7: Typical shape of the frequency response of a piezoelectric sensor. Zoom:

The usable frequency range is the flat region. Often the sensor will be connected to an electrical lowpass filter to attenuate the resonance peak.

Accelerometers

An accelerometer is a sensor that measures acceleration. Note that gravity is indistinguishable from acceleration, so accelerometers also detect the force of gravity. However, piezoelectric sensors are AC coupled, so a piezoelectric accelerometer inherently filters out the influence of gravity if the sensor is not rotating.

The two most common types of accelerometers are piezoelectric and MEMS (which are capacitive). Piezoelectric accelerometers can operate at higher frequencies but are more expensive than MEMS devices.

A typical piezoelectric accelerometer has a known mass supported by a piezo crystal. When acceleration occurs, a force $F=ma$ is applied to the crystal. The piezoelectric effect then converts the force into a voltage.

Often an amplifying circuit is embedded within the sensor to read out the voltage from the piezo crystal, since if there were a long cable between the crystal and the amplifier then there would be major challenges of noise, parasitic capacitance, etc.

Generally the sensor is only sensitive to movement along a single axis. To make a 3-axis accelerometer, three separate piezo crystals are used.

There is a widely adopted standard for the amplifier circuit called IEPE (integrated electronics piezoelectric). IEPE allows a single coax cable to simultaneously power the sensor and provide a voltage signal back to the data acquisition unit. A mental model of IEPE is shown in Figure 8.

Figure 8: Equivalent circuit of IEPE accelerometer from the perspective of the power supply. Zoom:

IEPE sensors are supplied with a constant current (typically 4 mA, although this varies). The circuit inside the sensor presents a variable impedance (e.g. by controlling a transistor) so that the voltage $V$ will vary. The maximum value of $V$ is typically 24 - 30 V. IEPE excitation sources are found in commercial data acquisition devices or provided as standalone devices with a simple voltage readout. You can also find reference designs from major component suppliers if you need to integrate this functionality into your own designs.

References

R.S. Dahiya, M. Valle, Robotic Tactile Sensing, Springer (2013), Appendix A.