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EuPhO 2024, Експеримент — Задача 2. Task E.2 - Piezoelectric properties

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"EuPhO 2024, Kutaisi, Georgia — Experimental Problems" · 10 т.

Условие

Some crystalline, electrically insulating materials, such as quartz and lead zirconate titanate, exhibit an electrical response when mechanical pressure is applied. In short, mechanical stress polarizes the crystals, which is called the piezoelectric effect. This phenomenon can be explained by the special structure of their molecules: the deformation gives each molecule an electric dipole moment. Conversely, mechanical stress is generated in the presence of an electric field, a phenomenon known as the reverse piezoelectric effect. However, in this problem, we will neglect the reverse piezoelectric effect.

This problem investigates a simple device that relies on piezoelectricity, the piezoelectric element. It consists of the piezoelectric material placed between two circular metallic plates. When a force, perpendicular to the metallic plates, is applied to the element, a force-dependent voltage appears between the plates.

Equipment (see also Fig. 1)

A Piezo element with electrical leads, attached to a wooden base plate and a cover plate with a small hole. The hole goes through the entire cover plate, such that a small part of the top electrode of the piezo element is visible through it. This electrode is thin and flexible.
B Multimeter (inner resistances are given, refer to the important remarks on the next page).
C 1.5 V AA battery with connector.
D Capacitor of unknown capacitance, with a diode soldered to one leg (when a forward current flows through the diode, the voltage drop across it is 0.56 V).
E 4 electric push-button switches (connected while pressed) on wires.
F 6 crocodile clamps.
G Digital scales (up to 10 kg).
H Digital stop watch.
I 2 different rubber balls.
J Wooden stand with adjustable release mechanism.
K 50 cm ruler.
L Small, wooden sticks (diameter of 2 mm), in two different lengths.
M Large metal screw.
N Wooden clothes peg.
O Pencil, pen and pencil sharpener.

Important practical remarks

It is expected that you provide circuit diagrams with all the electrical measurements you take. Use the symbols provided in Fig. 2.
Be careful not to short the multimeter! The internal resistances given are only valid for DC voltage measurements.
Do not exceed a total load of 100 N of force on the piezo element.
To connect two wires, it is recommended to wrap them around each other and use the crocodile clamps to secure the connection, see Fig. 3a.
The diode's polarity is visible in Fig. 3b.
You can adjust the height of the release mechanism via the two screws on the back, see Fig. 4. You can also adjust the horizontal position by unscrewing the vertical nut at the release mechanism.
Be careful not to let the bouncy ball escape. You can minimize the chances of the ball leaving your table by placing the stand with the release mechanism against one of the walls or in a corner of your desk and use the ruler as another, transparent wall that is secured with the clothes peg. If you still loose the ball, ask for assistance, do not leave your place.
After a certain time of inactivity, the multimeter will start to beep and, after another few minutes, will shut down. You can prevent shut-down by pressing any button.
Be aware that some regions of your table may spring back under load. It is wise to use stable regions of your table for the bouncing experiment.
For the evaluation of a series of measurements, you are expected to plot the data.
Only those measurements and evaluation methods that promise the highest precision and accuracy are awarded full marks. Choose your approaches accordingly while keeping in mind the precision of your tools. However, no marks are awarded for error estimation.
When measuring DC voltages with the multimeter in a range with the smallest digit resolution being $\delta U$ , the uncertainty $\Delta U$ of the measured value $U$ can be estimated via:

\Delta U = 0.7\,\% \cdot U + 3 \cdot \delta U\,. \qquad (3)

Photograph of two wires wrapped around each other, secured with a crocodile clamp. — Figure 3: Helpful remarks on the electric components. (a) Recommended electrical connection with wires wrapped around another and secured with a crocodile clamp.

Photograph of a black capacitor with a diode soldered to one leg; the circuit symbol of the diode is drawn beside it showing the correct orientation. — (b) The diode attached to the capacitor with the circuit symbol placed next to the diode in correct orientation.

The internal resistance of your multimeter in the DC voltmeter setting depends on the measurement range and model number. In Fig. 5, you can see where to find the model number of your voltmeter. The following table provides the internal resistances for two measurement ranges and all models.

Photograph of a red Uni-T multimeter; a red arrow points to the model number printed at the top right of the display. — Figure 5: The red arrow points towards the model number of the multimeter.

model number(s)	range	internal resistance
all models	2 V	11.1 M $\Omega$
UT33B+	200 mV	9.65 M $\Omega$
UT33C+	200 mV	9.91 M $\Omega$
UT33D+	200 mV	9.70 M $\Omega$

a Measure the capacitance $C$ of the capacitor (Fig. 3b). [2 т.]

b The metallic plates on the sides of the piezo element also act as a capacitor. Find the capacitance $C_p$ of the piezo element. [2,5 т.]

c Measure and plot how the voltage between the plates of the piezo element depends on the total perpendicular force, which is evenly distributed over the surface of the piezo via its wooden cover plate. For low forces, the dependence is linear; find the slope $\beta$ in this regime. [4 т.]

d The molecules of the crystals can only have polarizations lower than a certain critical value. Find the maximal (saturation) voltage of the piezo, the pressure $p_{sat}$ at saturation and the maximal surface density $\sigma_{max}$ of the charge on the surface of the piezo element. [1,5 т.]

Решение

Experiment 2024 - General Grading Scheme

General remarks for grading

For minor mistakes (like minus-sign, factor-2 wrong or just the inverse of a result), we deduct only $-0.1$ points once, and will grade the remaining task as if the mistake was corrected (no penalty for a propagating error), including the numerical results.
Even if the data used for making a plot is of low quality because of an inferior measurement technique, the points designated for plotting are not reduced as long as the data points are marked correctly according to the tabulated data.

Task E.2 - Piezoelectric properties

a) Capacitance of the capacitor We charge the capacitor with the battery, connect it to the multimeter in the 2V DC voltage range and switch off the battery using the top switch in the figure 2. If the bottom switch is not present, we start recording the time with the stopwatch the moment the switch is disconnected. If the bottom switch is present, we first disconnect the top switch, and start measuring time once the bottom switch is connected

Circuit diagram: voltmeter V on the left, capacitor C in the middle, battery U_0 (+ on top, − below) on the right; one switch in the top wire and one switch in the bottom wire. — Figure 2

The capacitor now discharges via the inner resistance ( $R_V = 11.1\,\mathrm{M\Omega}$ ) of the voltmeter. Every 5 seconds, we write down the voltage reading, see Tab. 2. The discharge of the capacitor can be described as:

\ln\frac{U(t)}{U_0} = -\frac{t}{R_V C} \qquad (2)

We plot the data, fit it to a linear function and compute $C$ from the slope (Fig. 3). We finally obtain a capacitance of:

C = (2.24 \pm 0.08)\,\mathrm{\mu F} \qquad (3)

Plot of ln(U(t)/U_0) / a.u. (0.0 to about −2.85) versus time / s (0 to 70): red diamonds as experimental values lying on a dark fitted straight line; legend 'fitted, C = ( 2.24 ± 0.08) μF' and 'experimental values'. — Figure 3

Circuit diagram for task E.2b: battery U_0 on the left, piezo element C_p, capacitor C, a discharge switch across C and a voltmeter V reading U_C on the right; the top wire carries three switches — between the battery and C_p, between C_p and C, and before the voltmeter. — Figure 4: Circuit diagram for task E.2b

Plot of U_C / mV (0 to about 420) versus N (0 to 30): experimental values with error bars and a red fitted straight line; legend 'fitted, γ = ( 14.77 ± 0.8) mV' and 'experimental values'. The points fall below the line for large N. — Figure 5

E.2a - Capacitance

		Points
a	Aproach and Measurement	0.9
	Idea to use the inner resistance of the voltmeter to discharge (documented via circuit diagram!)	0.3
	Exponential decay of the capacitor voltage expressed equivalently to Eq. 2	0.2
	0.1 per measured 20 s, up to 40 s	0.2
	0.1 per 4 data points (voltage, time) up to 8 total data points	0.2
b	Evaluation and result	1.1
	A graphical evaluation, or a fit via a calculator. Alternatively, a point-wise averages	0.5 or 0.3
	value of $C$ within $(2.28 \pm 0.08)\,\mathrm{\mu F}$ OR value of $C$ within $(2.28 \pm 0.12)\,\mathrm{\mu F}$	0.6 or 0.3
	Total on E.2a	2.0

Special cases and penalties

The reference capacitance in this grading scheme is the average of ca 20 capacitors and therefore differs slightly from Eq.(3).
It is also possible to calculate the total charge $q = \int V/R\,\mathrm{d}t$ leaving the capacitor during the discharge. In this case, the table above can be still used (0.2 will be awarded for $q = \int V/R\,\mathrm{d}t$ , instead of Eq. 2), and numerical integration substitutes the graphical evaluation.

b) Capacitance of the piezo It can be noticed that the capacitance of the piezo element is much smaller than $C$ , since with the same measurement technique as in E.2a, the piezo discharges too fast to take measurements of the direct discharging behaviour. However, with the provided switches and the capacitor, one can build a circuit (Fig. 4) which enables us to go through multiple cycles of charging the piezo with the battery and then discharging it onto the much larger capacitance $C$ .

After a number of cycles (above 10), the voltmeter can be connected via a switch to measure the voltage of the capacitor $U_C$ . After each voltage measurement, it is important to discharge both $C$ and $C_p$ , which can be done by two more switches in the circuit. We can now plot the capacitor voltage as a function of charging cycles (Fig. 5).

Detailed derivation After many charging cycles, the voltage across the capacitor approaches the voltage used to charge the piezo. With each cycle, the amount of charge transferred to the capacitor gets smaller.

Let us work out the charge transferred to the capacitor during the $N^{\mathrm{th}}$ charging cycle. Suppose it initially has voltage $U_{N-1}$ across it, meaning it has stored charge $Q_{N-1} = C U_{N-1}$ . Now connect the fully charged piezo, which has stored charge $Q_p = C_p U_0$ . After connecting them together, the combined system acts as a single capacitor with stored charge $Q_{N-1} + Q_p$ and capacitance $C + C_p$ , so the new voltage across the capacitor and the piezo is

U_N = \frac{Q_{N-1} + Q_p}{C + C_p} = \frac{C}{C+C_p}\,U_{N-1} + \frac{C_p}{C+C_p}\,U_0\,. \qquad (4)

When the voltage saturates, the change in one charging cycle is negligible so $U_N = U_{N-1}$ . In this case, (4) gives $U_N = U_{N-1} = U_0$ ; the voltage saturates at the voltage used to charge the piezo.

Suppose we are far from saturation, meaning $U_N \ll U_0$ . Rewrite (4) as follows:

\frac{C}{C+C_p}(U_N - U_{N-1}) + \frac{C_p}{C+C_p}\,U_N = \frac{C_p}{C+C_p}\,U_0\,. \qquad (5)

The condition $U_N \ll U_0$ means we can approximate this as

\frac{C}{C+C_p}(U_N - U_{N-1}) = \frac{C_p}{C+C_p}\,U_0\,,

which has a simple linear solution

U_N = N\,\frac{C_p U_0}{C}\,. \qquad (6)

This linear solution is a good approximation when $N C_p \ll C$ . We measure $U_0 \approx 1.405\,\mathrm{mV}$ . Fitting this linear relation can be used to measure $C_p$ . Here, the fit from Fig. 5 evaluates to

C_p = (23.5 \pm 2.4)\,\mathrm{nF} \qquad (7)

Note: the following precise calculations are not necessary for being awarded with full marks. The general solution to recursive equation (4) can be found as follows. Let $X_N = U_0 - U_N$ , which is the distance of $U_N$ from its saturation value. With this, (5) can be written as

X_N = \frac{C}{C+C_p}\,X_{N-1}\,.

The initial condition is $U_0 = 0$ , which means $X_0 = U_0$ . Then

X_N = \left(\frac{C}{C+C_p}\right)^N U_0\,.

Returning to the original variables, we arrive at

U_N = U_0 - \left(\frac{C}{C+C_p}\right)^N U_0\,. \qquad (8)

Indeed, in the limit $N \to \infty$ , this result gives the same saturation value that we found above.

Students are expected to fit the linear model $U_N \propto N$ . However, if they use (8) and fitting to

\log\left(\frac{U_0 - U_N}{U_0}\right) = N \log\left(\frac{C}{C+C_p}\right), \qquad (9)

they will also get full marks.

E.2b - Capacitance of the piezo

		Points
a	Aproach and Measurement	1.4
	Measurement technique to charge piezo (documented via circuit diagram!)	0.1
	discharge onto capacitor (documented via circuit diagram!)	0.2
	Idea to repeat multiple times to increase precision (multiple charging cycle idea)	0.4
	0.1 per measured 2 cycles, up to 6 cycles and after that 0.1 per measured 4 cycles up to 10	0.4
	0.1 per repetition per data point, up to 3 repetitions per point	0.3
b	Evaluation and result	1.1
	Approximation $N C_p \ll U_0$ leading to Eq. 6 and only taking into account values up to 15 charge cycles OR using Eq. 6 and only taking into account values up to 20 charge cycles OR exact solution like eq. 8 for arbitrary measurement range (also large cycle numbers allowed)	0.5 or 0.3 or 0.5
	value of $C_p$ inside $(23.5 \pm 3)\,\mathrm{nF}$ OR value of $C_p$ inside $(23.5 \pm 5)\,\mathrm{nF}$	0.6 or 0.3
	Total on E.2b	2.5

Special cases and penalties

$-0.1$ if $U_0$ was assumed to be 1.5 V instead of measuring it.
If no circuit diagram was provided, the first 0.3 points are deducted for the measurement technique and discharging the piezo on the capacitor.
Capacitor parallel to piezo case: This was quite common: If the student repeated the measurement from task E.2a but with the parallel circuit of piezo and capacitor (hence, measuring the sum of capacities), this is still a very imprecise measurement technique because the relative error will be on the order of the value that we seek to measure. It will however be possible to receive the following partial points if everything else was done as required: - 0.1 for measurement technique; - 0 for discharge on capacitor idea; - 0 for multiple charging cycles; - 0 for measured cycle number; - up to 0.3 for repetitions; - 0.2 for approach of using parallel capacitor; - max. 0.3 for correct value $\pm 5\,\mathrm{nF}$ range; - so, a maximum total of 0.9 points can be awarded.
Penalty: Student discharges piezo directly via Voltmeter If the student has only used a circuit that consists of the voltmeter and the piezo, (like in E.2a), the student always receives only the first 0.1 points for the entire task, given that they drew the circuit, else, 0 points.

c) Response function To measure the piezo voltage as a function of force, we assemble the circuit like shown in Fig. 6 and operate it as follows:

Circuit diagram: piezo element C_p with a discharge switch across it, a switch in the top wire to capacitor C, a discharge switch across C, and a switch in the top wire to the voltmeter V reading U_C. — Figure 6: .

Note that we need the capacitor, because if we were to connect the voltmeter directly to the piezo, it would discharge faster than we are able to take a reading. We place the piezo element on the scales and push down on the scales, while the switch connecting it to the capacitor is closed. When the desired weight is reached, we cut the capacitor off and release the weight from the scales. The voltage under load has been transferred to the capacitor and can now, via the voltmeter, be read out. Via $F = mg$ we can compute the force from the displayed mass values and by multiplying with $C/C_p$ , we obtain the piezo voltages from the capacitor voltages, using $C_p \ll C$ . We repeat the measurement 5 times per weight value to get an idea of fluctuations and utilize average values; the data can found in Table 4. The plot shown in Fig. 7 is achieved if we use most of the permitted force range of the scales.

Plot of U_p / V (0 to 120) versus F / N (0 to about 90): experimental values with error bars and a shaded band rising linearly and then saturating around 80–100 V; a red linear fit, a teal saturation fit with a shaded confidence band, and an orange vertical line at the saturation force with dashed lines around it. Legend: 'saturation force = 22.69 N', 'linear fit, β = ( 3.54 ± 0.08) V/N', 'fitted, β = ( 0.14 ± 0.28) V/N', 'experimental values'. — Figure 7

For large values of $F$ , the voltages become noisy and seem to saturate (see next sub-task). However, the dependence is fairly linear up to ca 10 N. We can fit that range and obtain a piezo voltage coefficient

\beta = (3.37 \pm 0.1)\,\mathrm{V/N} \qquad (10)

E.2c - Response function

		Points
a	Approach	0.9
	The idea to charge the capacitor using the piezo	0.4
	and measure afterwards the capacitor voltage with voltmeter	0.1
	using switch to discharge piezo (documented via circuit diagram or detailed description!)	0.4
b	Measurement	1.9
	If they did not discharge the piezo before each individual measurement, or if there are obvious points of piezo-zero floating, we half the measurement points	or 0.9
	0.05 for each measurement at a different weight (forces) value, up to for 16 data points; the result is to be rounded down to the precision of 0.1	0.8
	0.1 per data points within each 1 kg-wide weight range, up to 6 kg	0.6
	0.1 per repetition per data point (average number of repetitions for the entire measurement series), up to 5 repetitions per point	0.5
c	Evaluation and result	1.2
	Accurate plot of the data points	0.3
	Graphical evaluation of the slope, or fit with calculator of linear region OR point-wise average of linear region	0.4 or 0.2
	value of $\beta$ inside $(3.54 \pm 0.2)\,\mathrm{V/N}$ OR value of $\beta$ inside $(3.54 \pm 0.4)\,\mathrm{V/N}$	0.5 or 0.3
	Total on E.2c	4.0

Special cases and penalties

If no circuit diagram was provided, the first 0.9 points for the Approach are not awarded.
If the student uses only the voltmeter, to measure directly the piezo voltage, the results will not be reliable because the piezo discharges very quickly via the voltmeter, with the RC-time being equal to 20 ms. Therefore, this approach is very inaccurate, and no points (0/0.9) are given for the approach. For the measurements, only 50% of points can be given (0.9/1.9) points if an instantaneous voltage reading is taken after applying a force AND the result differs from the value that is expected for the given force no more than by a factor of 2.0. The points for Evaluation and result can be graded according to the table above.

d) Saturation If we already measured large enough weight values in the previous task, we can just use the data from there, otherwise, we record more data over a larger range now. Due to the noise in the saturated region, it is important to use multiple measurements per point and take the average. The saturation point can be found by computing the intersection of the fitting functions of the linear region for low forces and the saturated region, just as visible in Fig. 7. We can read out the relevant values, having measured the area of the piezo as $A_p = \pi r_p^2 \approx 4.15\,10^{-4}\,\mathrm{m^2}$ :

U_{sat} = (86 \pm 16)\,\mathrm{V} \qquad (11)

and

p_{sat} = F_{sat}/A_p = (55000 \pm 7000)\,\mathrm{Pa} \qquad (12)

and therefore,

\sigma_{max} = \frac{C_p U_{sat}}{A} = (4.9 \pm 0.9)\,\mathrm{mC/m^2} \qquad (13)

E.2d - Saturation

		Points
a	Aproach and Measurement	0.6
	Determination method of saturation point as intersection of extreme region fit functions (using data from Figure 7)	0.4
	Formula Eq. 13	0.2
b	Evaluation and result	0.9
	Graphical evaluation, or calculation of intersection	0.3
	value of $U_{sat}$ inside $(86 \pm 25)\,\mathrm{V}$	0.2
	value of $p_{sat}$ from 40 kPa to 85 kPa	0.2
	value of $\sigma_{max}$ inside $(4.9 \pm 2)\,\mathrm{mC/m^2}$	0.2
	Total on E.2d	1.5

Special cases and penalties

If measurements additional to what have been done for E.2c are made here, these are to be taken into account while grading E-2c-b.
Common case If the saturation was not determined via a plot but simply by pressing and ballparking the force at which the voltage does not rise anymore, we can award points up to the following maxima: - 0.2/0.4 for evaluation method (because it works in principle but can be very imprecise); - 0.2 if the formula was given; - 0 for graphical evaluation - at least a plot with meaningfull graphical interpretation (could be the one from E.2c) is required to correctly assess the situation here. Without (which is this common case) it's 0; - max $3 \times 0.2$ for all remaining values, graded just normally; - So a total of 1.0/1.5 points can be awarded for this technique.
Penalty: Without a way to quantitatively measure piezo voltage responses to applied forces in E.2c, (especially if E.2c was not done or measurements were only attempted with a voltmeter directly attached to the piezo without anything else), this task can not be done in a way that is physically meaningfull and it will always receive 0 points.

Оригинал в Архива: eupho2024_experiment_problems.pdf · официални решения: eupho2024_experiment_solutions.pdf