Съдържание

Условие Part A. Capacitors at room temperature (4.0 points)Part B. Calibrating NTC thermistor (1.0 point)Part C. Capacitors at different temperatures (3.0 points)Part D. Sources of measurement errors (2.0 points)Отговори Решение

IPhO 2021, Експеримент — Задача 1. Non-ideal capacitors

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Experiment — IPhO Lithuania 2021 — Q1 — Non-ideal capacitors (10 points) · 10 т.

Условие

This experiment is designed to investigate the properties of capacitors.

Capacitor’s capacitance (which always means differential capacitance in this text) can be found based on its charging graph of its voltage $U(t)$ via the resistor $R_1$ . Depending on the circuit, it is necessary to find the relation of capacitor’s charging current vs voltage $I(U)$ and use it to determine capacitance:

C(U) = \frac{\mathrm{d}q}{\mathrm{d}U} = \frac{I\mathrm{d}t}{\mathrm{d}U} = \frac{I\,(U)}{\mathrm{d}U/\mathrm{d}t}. \qquad (1)

The electric circuit implemented in this experiment is shown in Fig. 1.1. Switch S1 on the board can be used to switch between capacitors C1 and C2. The middle position of the switch does not play any role in this experiment and should never be used.

Circuit diagram. A shaded board holds capacitors C1 and C2, both grounded on one side, connected through the two-position switch S1 to a node joined to the Voltmeter and to terminal IN; the Voltmeter's other side goes to ground and terminal GND. Board terminals on the right: +9V, −9 V, IN, GND. Off the board, jumper wire W1 contains resistor R1 of 100 MΩ, and jumper wire W2 is a plain wire; both end in connectors that plug into the terminals. — Figure 1.1. Electric circuit for the experiment.

Caution: one of the sample capacitors contains a dielectric with dielectric permittivity that depends on the capacitor voltage change rate. To keep this rate as stable as possible, when measuring at the positive voltages, the capacitor should be charged from 9 V down to –9 V, while measurements at the negative voltages should be done when capacitor is charged from –9 V towards 9 V. The measured capacitance can be influenced by the previous state of the capacitor, thus capacitor should be kept at the starting voltage for at least 10 s before the measurement.

Part A. Capacitors at room temperature (4.0 points)

Measure and graph the capacitance of the capacitors C1 and C2 versus the voltage at room temperature (draw all graphs together on the same axes).

A.1 Measure and graph $C_1(U)$ and $C_2(U)$ in range from –7 V to 7 V. In the answer sheet write $C_1$ and $C_2$ values at 0 V, 3 V, and 6 V. Write down the formula used for calculating capacitance from raw measurements. Also write Board ID and room temperature. [2,3 т.]

A.2 Find the voltage $U_{\mathrm{max\ change}}$ , at which the capacitance of capacitor exhibits the fastest relative change versus the voltage $\left(\frac{\mathrm{d}C(U)}{C(U)\mathrm{d}U}\right)$ . In the answer sheet write, which capacitor (C1 or C2) exhibits the fastest change and the voltage, at which it is observed. [0,5 т.]

A.3 What are the charges $q_1$ and $q_2$ of capacitors C1 and C2 at 6 V? [1,2 т.]

Part B. Calibrating NTC thermistor (1.0 point)

Measure the NTC (Negative Temperature Coefficient) thermistor voltage at a known room temperature (from examination hall thermometer). The formula (1) for its resistance vs temperature and it’s circuit is show in ”Experimental Examination - Overall Guide G1”.

B.1 Find the NTC thermistor constant $R_0$ . [1 т.]

Part C. Capacitors at different temperatures (3.0 points)

C.1 Measure and graph $C_1(U)$ and $C_2(U)$ in range from –7 V to 7 V at temperatures of 40 °C, 65 °C and 85 °C. [1,3 т.]

C.2 Graph $C_1(T)$ and $C_2(T)$ at 0 V and 6 V versus temperature from room temperature up to 85 °C. [0,5 т.]

C.3 In the answer sheet write the ratio $C(85\ {}^{\circ}\mathrm{C})/C(40\ {}^{\circ}\mathrm{C})$ for both capacitors C1 and C2 at 0 V and 6 V. [1,2 т.]

Part D. Sources of measurement errors (2.0 points)

The previous tasks in this experiment were done in conditions of long initial charge. When looking at shorter recharging times (0.1 - 10 s) there can be multiple sources of errors:

Leakage current.
Polarization properties of the capacitor’s dielectric media that can be expressed as the dielectric permittivity that depends on process time scale.

Caution: heat-insulating material may absorb air moisture and become conductive. Remove it when doing leakage measurements.

Determine the main source of error for measuring C1 and C2, since capacitor leakage and voltmeter input currents depend on the voltage, estimate these errors at voltage close to 9 V. Decide, which auxiliary measurements and under what conditions should be taken in order to answer these questions. In your answers to the following D.1 and D.2 questions, you might indicate the conditions of your measurements, which quantities you measure and what conclusions you make based on your measurements, as exemplified in the tables below.

Note: these are just the examples how to describe schematically your measurements; you need determine the relevant conditions of your measurements by yourself.

Examples of how answers to questions D.1 and D.2 should be written:

Example 1.
Showing that voltage change rate of C1 connected to the measuring circuit is faster at 9 V than at 0 V.
Possible S1 positions: C1, C2
Possible IN connection: +9V, -9V, GND, Free
Initial settings:

S1 position	IN connection
C1	9V

Process:

Step number	S1 position	IN connection	Measured variable
1	C1	Free	$\lvert\mathrm{d}uC(t)\rvert/\mathrm{d}t$
2	C1	GND
3	C1	Free	$\lvert\mathrm{d}uC(t)\rvert/\mathrm{d}t$

Verification: $\lvert\mathrm{d}uC(t)\rvert/\mathrm{d}t\rvert_1 > \lvert\mathrm{d}uC(t)\rvert/\mathrm{d}t\rvert_3$

Example 2.
Showing that voltage change rate of C1 at 9 V is larger than the average voltage change rate starting at 0 V over 1000 seconds.
Possible S1 positions: C1, C2
Possible IN connection: +9V, -9V, GND, Free
Initial settings:

S1 position	IN connection
C1	9V

Process:

Step number	S1 position	IN connection	Duration, s	Measured variable
1	C1	Free		$\lvert\mathrm{d}uC(t)\rvert/\mathrm{d}t$
2	C1	GND
3	C1	Free		$uC$
4	C1	Free	1000
5	C1	Free		$uC$

Verification: $\lvert\mathrm{d}uC(t)\rvert/\mathrm{d}t\rvert_1 > (uC\rvert_3 - uC\rvert_5)/1000$

D.1 What is the main source of error for measuring $C_1(9\,\mathrm{V})$ ? Write the measurement steps in the tables. [1 т.]

D.2 What is the main source of error for measuring $C_2(9\,\mathrm{V})$ ? Write the measurement steps in the tables. [1 т.]

Отговори

Решение

Capacitance measurement method:

First, measure the highest voltage the capacitor can reach by connecting it to the voltage source via jumper wire W2. Before each measurement, connect capacitor to starting voltage source with jumper wire W2 and to a final voltage source ( $U_f$ ) with jumper wire W1 via the resistor R1. Capacitor C2 should be prepared that way for at least 10 s, while C1 measurement can be started immediately by disconnecting jumper wire W2 from the starting voltage source. To determine a precise value of the final voltage $U_f$ , it should be measured after capacitor has been connected to final source via R1 for a long time (at least 3 minutes). Then, the capacitance can be calculated from:

C(U) = \frac{U_f - U(t)}{R1}\Big/\frac{\mathrm{d}U}{\mathrm{d}t}

When measuring C2, to ensure minimal change in charging current, capacitance should only be calculated in conditions where $U_f$ and $U(t)$ have different polarities. This way, capacitance dependence on voltage should be symmetrical around 0 V.

Part A: Capacitors at room temperature (4 points)

A.1 (2.3 pt)

Graph $C_1(U)$ should be constant, $C_2(U)$ must be highest at 0 V.
Example results measured at room temperature of 29 °C.

	$C_1$	$C_2$
0 V	0.100 μF	0.473 μF
3 V	0.100 μF	0.183 μF
6 V	0.100 μF	0.086 μF

C(U) = \frac{U_f - U(t)}{R1}\Big/\frac{\mathrm{d}U}{\mathrm{d}t}.

A.2 (0.5 pt)

$U_{\mathrm{max\,change}} = 1.6\ \mathrm{V}$ at capacitor C2

A.3 (1.2 pt)

It’s important to calculate $\int_{0V}^{6V} C(U)dU$ , not just attempt to multiply $C(6\ \mathrm{V})\cdot 6\ \mathrm{V}$

q_1 = 0.60\ \mu\mathrm{C};\quad q_2 = 1.3\ \mu\mathrm{C}

Part B: Calibrating NCT thermistor (1 point)

B.1 (1.0 pt)

$R_0 = \frac{U_{T0}R_3}{U-U_{T0}}\mathrm{e}^{-B/T}$ , where $U = 3.3\ \mathrm{V}$ , $U_{T0}$ – $uT$ at room temperature, $T$ – room temperature in kelvins

$R_0 = 0.0341\ \Omega$ .

Part C: Capacitors at different temperatures (3 points)

C.1 (1.3 pt)

Graphs $C_1(U,T)$ should always stay constant, $C_2(U)$ must be highest at 0 V

C.2 (0.5 pt)

Graph $C_1(T)$ should always stay constant

C.3 (1.2 pt)

$C_1(85\ {}^{\circ}\mathrm{C})/C_1(40\ {}^{\circ}\mathrm{C})\ \rvert_{0\mathrm{V}} = 1.00$

$C_1(85\ {}^{\circ}\mathrm{C})/C_1(40\ {}^{\circ}\mathrm{C})\ \rvert_{6\mathrm{V}} = 1.00$

$C_2(85\ {}^{\circ}\mathrm{C})/C_2(40\ {}^{\circ}\mathrm{C})\ \rvert_{0\mathrm{V}} = 0.63$

$C_2(85\ {}^{\circ}\mathrm{C})/C_2(40\ {}^{\circ}\mathrm{C})\ \rvert_{6\mathrm{V}} = 1.06$

Part D: Sources of measurement errors (2 points)

D.1 (1.0 pt)

Initial settings:

S1 position	IN connection
C1	-9V or GND

Process:

Step number	S1 position	IN connection	Duration, s	Measured variable
1	C1	+9V	0.2 s (any short time is good)
2	C1	Free		$\lvert\mathrm{d}uC(t)\rvert/\mathrm{d}t$
3	C1	+9V	5 s (has to be much longer than first)
4	C1	Free		$\lvert\mathrm{d}uC(t)\rvert/\mathrm{d}t$

Verification: $\lvert\mathrm{d}uC(t)\rvert/\mathrm{d}t\rvert_2 = \lvert\mathrm{d}uC(t)\rvert/\mathrm{d}t\rvert_4$

Main source of error: 1 (Leakage current.)

D.2 (1.0 pt)

Initial settings:

S1 position	IN connection
C2	-9V or GND

Process:

Step number	S1 position	IN connection	Duration, s	Measured variable
1	C2	+9V	0.2 s (any short time is good)
2	C2	Free		$\lvert\mathrm{d}uC(t)\rvert/\mathrm{d}t$
3	C2	+9V	5 s (has to be much longer than first)
4	C2	Free		$\lvert\mathrm{d}uC(t)\rvert/\mathrm{d}t$

Verification: $\lvert\mathrm{d}uC(t)\rvert/\mathrm{d}t\rvert_2 \gg \lvert\mathrm{d}uC(t)\rvert/\mathrm{d}t\rvert_4$

Alternatively,

\frac{\lvert\mathrm{d}uC(t)\rvert/\mathrm{d}t\rvert_2}{\lvert\mathrm{d}uC(t)\rvert/\mathrm{d}t\rvert_4} > 2.

Main source of error: 2 (Polarization properties of the capacitor’s dielectric media)

Graph of C, μF (0.0 to 0.5) versus U, V (−7 to 7). Blue circles C1 form a flat line at about 0.1 μF; orange triangles C2 form a symmetric peak of about 0.45 μF at 0 V, falling to about 0.08 μF at ±7 V and crossing C1 near ±5 V. Legend: C1, C2. — A.1: example results measured at room temperature of 29 °C.

Graph of C, μF (0.0 to 0.5) versus U, V (−7 to 7) at three temperatures. C1 at 40 °C, 65 °C and 85 °C (circles) is flat at about 0.1 μF; C2 (triangles) peaks at 0 V at about 0.40 μF (40 °C, blue), 0.30 μF (65 °C, orange) and 0.25 μF (85 °C, green), converging near 0.08–0.1 μF at ±5 to ±7 V. Legend: C1 40 °C, C2 40 °C, C1 65 °C, C2 65 °C, C1 85 °C, C2 85 °C. — C.1

Scatter plot of C, μF (0.0 to 0.5) versus T, °C (about 27 to 87). C2 at 0 V (orange triangles) decreases from about 0.46 μF at 29 °C to 0.40 at 40 °C, 0.30 at 65 °C and 0.25 at 85 °C; C1 at 0 V (blue circles), C1 at 6 V (green squares) and C2 at 6 V (red down-triangles) all lie near 0.09–0.1 μF at every temperature. Legend: C1 0 V, C2 0 V, C1 6 V, C2 6 V. — C.2

Оригинал в Архива: IPhO 2021 Exp Q1.pdf · официални решения: IPhO 2021 Exp S1.pdf