APhO 2015, Експеримент — Experiment Problem. The Piezoelectric Effect And Its Applications

Experiment A — Basic measurement (3 т.)

In this experiment, you are required to measure the dimensions, the mass and the capacitance of a PZT plate, and then calculate its density $\rho$ and relative permittivity $\varepsilon_r$.

Please choose a PZT plate. You are supposed to perform Experiments A, B and C using this same plate.

In Experiments A to E, error analysis is required if it is explicitly stated; it is not required if it is not stated.

A.1 Choose a PZT plate and use the Vernier caliper to measure its length $l$, width $w$ and thickness $t$. Use the electronic weighing scale to measure its mass $m$. Use the DMM and the Kelvin clip to measure its capacitance $C$ (at ambient temperature).

Considering the slight non-uniformity in the dimensions of the PZT plate and the uncertainties of instrumental readings, repeat each measurement several times and then calculate the mean and the standard error. [1,6 т.]

Attention: The relative permittivity of the PZT plate is temperature dependent (see Experiment C). You are supposed to perform the capacitance measurement at ambient temperature. Avoid the direct warming up of the plate by your hand.

A.2 Now calculate the density $\rho$ and the relative permittivity $\varepsilon_r$ of the PZT plate. Based on standard errors obtained from A.1, carry out the error analysis to estimate the uncertainties of $\rho$ and $\varepsilon_r$ (vacuum permittivity $\varepsilon_0=8.85\times10^{-12}\ \mathrm{F/m}$). [1,4 т.]

Experiment B — The resonant method to measure the piezoelectric coefficient (4,5 т.)

As described in the introduction section, the piezoelectric plate produces distortion (also called strain $S$) when subjected to an electric field. The proportional coefficient $d$ of the strain $S$ versus the electric field strength $E$ is defined as the piezoelectric coefficient

In reality, the PZT plate is anisotropic. There is a special direction called the polarization direction. During production of the PZT plate, a strong DC electric field is applied along its thickness direction ($z$-axis in Figure 7) to align the molecular dipoles of the ceramic at a temperature higher than the Curie temperature (see Experiment C). This polarization remains after temperature is reduced below the Curie temperature and then the DC electric field is removed.

The top and bottom flat surfaces of the plate are coated with silver films as electrodes (see Figure 7). The electric field is along $z$-axis (3) when the electrodes are connected to a voltage source, and we shall denote it as $E_3$. Here we define

where $S_1=\Delta l/l$ and $S_3=\Delta t/t$ are strains along $x$-axis (1) and $z$-axis (3), respectively. Note that the strain is not necessarily parallel to the electric field $E_3$. For PZT materials, $d_{31}$ is roughly half of $d_{33}$. According to the parameters in Experiment A, it can be shown that length $l$ changes the most when a voltage $V$ is applied across the electrodes, i.e.,

where $l/t\gg w/t\gg2$. To simplify the theoretical treatment, for such a long thin plate, vibrations along the width ($y$-axis) and thickness ($z$-axis) directions can be neglected and the problem reduces to a one-dimensional vibration problem. As such we remove the redundant subscript and simply denote $d_{31}$ as $d$. In Experiments D and E the ignored vibration along the width ($y$-axis) direction may cause slight imperfection to the measurement.

The PZT plate performs like a pure capacitor (with a capacitance $C$ from A.1) when driven by low-frequency signals. However, as frequency increases, the vibration of the PZT plate changes its circuitry behavior significantly. At certain frequencies, called resonant frequencies, the plate vibrates strongly and its impedance reaches a minimum. Along with the resonant frequencies, there are also frequencies where the impedance reaches a maximum, and we call them antiresonant frequencies.

The first resonant frequency $f_r$ of the plate is associated with its fundamental vibration mode along the length direction ($x$-axis). Near $f_r$, the PZT plate can be approximated by a simple circuit, with two capacitors ($C_0$ and $C_1$) and an inductor ($L_1$) being arranged as shown in Figure 8.

B.1 Derive the expressions for the resonant frequency $f_r$ and the antiresonant frequency $f_a$ of the equivalent circuit. [1 т.]

The piezoelectric coefficient $d$ can be calculated by the following formula

Now we perform the experiment to locate $f_r$ and $f_a$. See the circuit schematics in Figure 9. We keep the output amplitude (the voltage $V$) of the signal generator constant, such that the impedance of the PZT plate correlates with the AC current in the circuit.

B.2 Measure the AC current $I$ through the PZT plate as a function of the signal frequency $f$. Draw the $I$-$f$ curve and find the resonant frequency $f_r$ and the antiresonant frequency $f_a$. Calculate the piezoelectric coefficient $d$ accordingly. [3,5 т.]

Instructions:

(1) Connect the signal generator, the DMM and the PZT plate according to Figure 9. Note that the PZT plate should be clamped at the middle point along the length direction using the Kelvin clip, since the middle point is the node under the free boundary condition.

(2) The amplitude of the output signal changes little with frequency if the “Amplitude” knob of the signal generator remains untouched. However, the DMM may not respond to signals with a frequency above 40 kHz. Data above 40 kHz are not required.

(3) Attention: keep the clamped PZT plate intact during the experiment. Slight movement may cause drifting of the current reading.

(4) Attention: as mentioned in A.1, you are supposed to perform the measurement at ambient temperature. Avoid the direct warming up of the plate by your hand.

(5) Attention: if you do not hear any high pitch sound from your testing plate after sweeping frequency and you are confident with your measurement setup, you should contact the organizer as your instruments might be faulty.

Experiment C — The Curie temperature of the PZT plate (4 т.)

Most insulators have a dielectric susceptibility which is insensitive to the temperature variation. However, the static relative permittivity of the PZT ceramics changes with temperature according to the relation

Here $A$ and $B$ are constants independent of temperature. This relation is known as the Curie-Weiss law. Parameters $B$ and $T_c$ are called the Curie constant and the Curie temperature respectively, named after Pierre Curie.

A phase transition occurs at the transition temperature, $T_c$. Above $T_c$, the substance is in the paraelectric phase, in which the elementary dipoles of the various unit cells in the crystal are oriented randomly. Below $T_c$, the elementary dipoles interact with each other, giving rise to an internal field, which lines up the dipoles. A spontaneous polarization appears in the absence of an applied electric field. The relative permittivity below $T_c$ is given as

From Experiment A we know that $\varepsilon_r\gg1$. We can ignore the constant term to obtain an approximation of the static relative permittivity as

Therefore capacitance of the PZT plate obtained from A.1 could vary with temperature as well. Since the Curie temperature of the PZT plate is much higher than the boiling temperature of water, we will estimate its Curie temperature by linear extrapolation.

C.1 Now measure the capacitance of the PZT plate at various temperatures and record the data. [1,5 т.]

Instructions:

(1) Use the cable with two banana plugs and two crocodile clips to connect the PZT plate to the DMM. Pay attention to the polarity of the two crocodile clips when clamping the PZT plate. Do not use the Kelvin clip because its ABS plastics will soften when heated.

(2) Put the PZT plate inside the plastic bag. Use the plastic clip to clamp the cable and the bag. Warning: one side of the bag is open. Do not tear the other side!

(3) Pour the 1.5 L bottled water into the water bath. Immerse the plastic bag in the water. Use the pebble to keep the plastic bag in the water.

(4) Turn on the water bath and set the target temperature. Warning: it is strictly prohibited to turn on the machine if the heating unit is not covered by water.

(5) Warning: be very careful with the hot water in the water bath. Remember that water at temperatures higher than $50^\circ\mathrm{C}$ can cause burns. For safety please do not set the temperature higher than $90^\circ\mathrm{C}$.

(6) The temperature will rise slowly. Please record the capacitance $C$ of the PZT plate at different temperatures.

(7) Turn off the water bath and unplug the power cord from the power socket when the measurement is finished.

(8) Hint: to speed up the experiment, you can choose to set the target temperature to $90^\circ\mathrm{C}$ and record the capacitance as the temperature rises.

C.2 Analyze the data, draw a proper plot and calculate the Curie temperature accordingly. [2,5 т.]

Experiment D — Application: measuring the speed of sound in aluminum (6,5 т.)

In solids, sound can be transmitted as both longitudinal waves and transverse waves. The medium movements responsible for the two types of waves are illustrated below.

In longitudinal waves, oscillation occurs in the longitudinal direction or the direction of wave propagation. Longitudinal waves can be generated in liquids, as well as in solids because the energy travels through the atomic structure by a series of compression and expansion movements.

In transverse waves, displacement of the medium is perpendicular to the direction of wave propagation. Transverse waves require a solid medium for propagation, and cannot effectively propagate in liquids or gasses.

In this experiment, we will measure the longitudinal and transverse wave velocities of sound in an aluminum rod. We will first measure the transverse wave velocity, and then the longitudinal wave velocity.

D.1 Assume that the length of the aluminum rod is $L$ and the wave velocity is $u$. Under the free boundary condition, derive the equation for the frequencies $f_n$ of the standing (resonant) waves along the long rod. Then derive the equation for the wave velocity $u$ from $f_n$. [0,6 т.]

Now, we use a PZT plate as a transducer to produce sound waves in the aluminum rod, and use another PZT plate as a vibration sensor to detect the reflected sound waves.

First we measure the transverse wave velocity. As discussed in Experiment B, the vibration along the length direction is dominant. We position the transducer and the sensor to one end of the rod, as illustrated in Figure 11. The vibration of the transducer will propagate into the rod via friction, forming transverse waves.

D.2 Use the steel tape measure to read the length $L$ of the aluminum rod. Please repeat the measurement several times and calculate the mean and the standard error.

While changing the frequency of the sound waves produced by the transducer, record the peak values monitored by the sensor. Draw a spectrum containing all measured resonant peaks, similar to that shown in Figure 12. [1,6 т.]

Instructions:

(1) Following steps 2 to 4 to arrange the experimental setup as illustrated in Figure 11. Note that the plastic box is specifically designed to accommodate the setup.

(2) Place the aluminum rod in the long trough of the plastic box, and use the eraser to press the PZT plates against one end of the aluminum rod, as illustrated in Figure 11. The two plates should not touch each other.

(3) Put the spring on the other end of the trough to gently push the rod against the PZT plates.

(4) Connect one PZT plate to the signal generator as the transducer. Connect the other PZT plate to the DMM as the sensor. Pay attention to the polarity when using the crocodile clips.

(5) The PZT plate is fragile and there is no extra supply.

(6) Use earplugs if you are annoyed by the high pitch sound.

(7) It is recommended that you sweep the frequency in the range between 0 and 40 kHz.

D.3 Identify the resonant peaks likely resulting from the transverse waves. Calculate the transverse wave velocity accordingly and carry out the error analysis.

Attention: there might be irrelevant peaks caused by imperfection of the experimental setup, e.g., imperfect free boundary condition. You need to make a judgement and ignore the irrelevant peaks during your analysis. [1,4 т.]

By changing the contact style between the PZT plates and the rod, we can also measure the longitudinal wave velocity in the rod. Attach the transducer and the sensor to the rod as illustrated in Figure 13. Vibration along the length direction of the transducer will propagate into the rod via compression, forming longitudinal waves.

As discussed in Experiment B, although vibration along the length direction is dominant, vibrations along other directions also exist, which could produce sound waves in the rod and introduce extra resonant peaks in the spectrum as well. Note that vibrations along other directions only introduce transverse waves. Therefore the extra resonant peaks will not coincide with the peaks caused by the longitudinal standing waves.

D.4 While changing the frequency of the sound waves produced by the transducer, record the peak values monitored by the sensor. Draw a spectrum containing all measured resonant peaks, similar to that shown in Figure 12. [1,5 т.]

Instructions:

(1) Following steps 2 to 5 to arrange the experimental setup as illustrated in Figure 13. Note that the plastic box is specifically designed to accommodate the setup.

(2) Align two PZT plates in the two slim slots, respectively, on one end of the plastic box.

(3) With the long aluminum rod in the trough, use the eraser as a cushion to press the PZT plates against one end of the rod.

(4) Insert the spring on the other end to gently push the rod against the PZT plates.

(5) Connect one PZT plate to the signal generator as the transducer. Connect the other PZT plate to the DMM as the sensor. Note the polarity when using the crocodile clips.

(6) Attention: the style of contact between the PZT plates and the rod is crucial. Please have the edges of the PZT plates fully in touch with the side of the rod, so as to avoid point or partial contact.

(7) Attention: if you find too many resonant peaks in the spectrum, you can try to reduce the signal amplitude from the signal generator and/or slightly release the pressure between the PZT plates and the rod to reduce the production of transverse waves. You can also try to remove the spring on the other end if the contact between the PZT plates and the rod is still decent without the spring.

(8) It is recommended that you sweep the frequency between 0 and 40 kHz.

D.5 Compare with the result in D.2, identify the resonant peaks caused by the transverse waves. Select the resonant peaks resulting from the longitudinal waves and calculate the longitudinal wave velocity accordingly. Carry out the error analysis. [1,4 т.]

Experiment E — Application: locating a defect in an aluminum rod (2 т.)

After knowing the longitudinal wave velocity from Experiment D, here we attempt to locate an artificially-engineered defect, a deep cut, along the long aluminum rod.

Sound waves through solid materials have been used to detect hidden cracks, voids, and other internal discontinuities in metals, composites, plastics, and ceramics. The industrial ultrasonic testing is done by repeatedly generating a several megahertz short pulse with an amplitude up to several hundred volts to drive the transducer, and then by amplifying the received signal to see if there is any reflection from a flaw. This is technically too complicated for our current setup. Instead, as a simple demonstration, we use the resonant method again to detect a deep, flat cut in the rod. Different from the case in Experiment D, sound waves will be bounced back at the spot of the deep cut instead of the far end of the rod. Therefore arrangement of the resonant peaks in the spectrum will be different from that in Experiment D, which can be the criterion for locating the spot of the deep cut.

The detection can be done using longitudinal waves.

E.1 While changing the frequency of the sound waves produced by the transducer, record the peak values monitored by the sensor. Draw a spectrum containing all measured resonant peaks, similar to that shown in Figure 12. [1,2 т.]

E.2 In the measured spectrum, identify the resonant peaks corresponding to the existence of the deep cut. Estimate the distance from the spot of the cut to the end of the rod that is in contact with the PZT plates. [0,8 т.]

Answer Sheets

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