APhO 2012, Експеримент — Задача 2. ELECTROMAGNETIC INDUCTION

13th Asian Physics Olympiad — India — Experimental Competition — Wednesday, 2nd May 2012 · 2 май 2012 г.

Условие

The modern technique of eddy current testing employed for detecting defects under the surfaces of metallic objects is based on the principles of electromagnetic induction. The circulating currents induced in conducting bodies due to changing magnetic flux in the region where they are located are called eddy currents. The defects are detected by observing the changes in the resistance and inductance of a coil carrying alternating current when held near the surface of the object.

Unless the core of the solenoid has some ferromagnetic material the magnetic flux $\phi$ is proportional to the current $i$ .The constant of proportionality is called self inductance of the coil and is represented by letter $L$. The self-induced emf in a coil with inductance $L$ is, therefore, given by $-L\dfrac{di}{dt}$.

In a coupled system of circuits with current $i_1$ in one circuit the magnetic flux linked with it is given by

Where $L_1$ is the self inductance of that circuit and $M_{21}$ is the mutual inductance of the coupled system.

Similar equation holds for the reverse, with $M_{12}=M_{21}$

L-R Circuit

A sinusoidal alternating current with angular frequency $\omega$ flowing through a series combination of resistance $R$ and inductance $L$ produces a voltage drop across the combination.

If we represent current by $i = I_0 \sin \omega t$, the voltage drop across the resistance is equal to $I_0 R \sin \omega t$ and that across the inductance it is $I_0 \omega L \cos \omega t$. We can combine these to get the voltage across the $R$-$L$ combination. The quantity $\omega L$ is called inductive reactance and is represented by symbol $X$. One can readily show that the voltage across the coil is equal to $I_0 Z \sin(\omega t+\theta)$, where

and

Alternating voltage as well as current vary continuously in both magnitude and polarity during the course of time but the rms values of these quantities calculated over a cycle are independent of time and the relation $V = IZ$ where both $V$ and $I$ represent the rms values, is analogous to Ohm’s law. From this we see that

(Note:- The concept of resistance is basically related to dissipation of electrical energy and the value of resistance of a coil in ac circuit can be different from its value determined by applying Ohm’s law with dc currents.)

When there are additional resistances and/or inductances in series with the coil, we can still consider the voltage drop across the combination as equal to the square root of the sum of the squares of voltage drops across the total resistance and across the total inductance.

Measurement of Inductance and Resistance of a Coil

For measuring alternating current and voltage, generally the rms values are noted. From Equations (2) & (3) we get

and

To obtain the values of $R$ and $L$ of the coil we can use the above equations. Voltage $V$ and current $I$ can be measured. But there being three unknown quantities $\theta$, $R$ and $X$ we need one more equation.

If the applied voltage across the series combination consisting of a known resistor $R'$ with the coil is $V_\mathrm{A}$, then an expression relating applied voltage $V_\mathrm{A}$ to the voltage drops $V_{R'}$ across $R'$, $V$ across the coil and the angle $\theta$ is

All quantities except $\theta$ in Equation (6) are measurable.

Hence measuring the three voltages $V_\mathrm{A}$, $V_{\mathrm{R}'}$ and $V$, and using Equations (4), (5) and (6), $\theta$, $R$ and $X$ can be determined. Knowing the frequency of the alternating current the value of $L$ can be calculated.

Alternatively, from Equations (4) and (6) we can express the value of $R$ in terms of the three measured voltages as

The impedance $Z$ of the coil can be calculated using the formula $Z=\dfrac{V}{V_{R'}}R'$ and the value of $X$ could be obtained from

Coupled Circuits

The energy supplied by the power source to the primary can be dissipated partially in the primary and the remainder in other mutually coupled secondaries. When no mechanical work is done, the energy dissipation is only in resistances. The inductances store energy in the magnetic field associated with them. With current $I$ (rms value) the average stored energy in inductance $L$ is equal to $\dfrac{1}{2}LI^2$.

When a current flows in the secondary the emf induced due to it in the primary brings about change in the primary current. Seen from the primary side the effect is a consequence of change in the effective resistance and reactance of the primary coil and there is no need to know the parameters in the secondary circuit. The total energy dissipated in primary as well as secondary circuits appears as if it is dissipated in the effective primary resistance when seen from the primary side.

The effective values of resistance $R_\mathrm{PE}$ and inductance $L_\mathrm{PE}$ of primary can be related to a ‘reflected’ resistance $R_\mathrm{R}$ and a ‘reflected’ inductance $L_\mathrm{R}$ from the secondary side. The (average) power dissipated in the reflected resistance $R_\mathrm{R}$ in the primary has to be equal to that in resistance $R_\mathrm{S}$ in the secondary circuit. This gives

Similarly, we can relate the reflected inductance $L_\mathrm{R}$ to the secondary inductance $L_\mathrm{S}$ from

Considering the fact that the induced emf in the secondary due to an alternating primary current $I_\mathrm{P}$ has magnitude equal to $\omega M I_\mathrm{P}$ we can write the equation corresponding to Kirchhoff’s loop rule for the secondary in terms of rms values of primary and secondary currents as

where $Z_\mathrm{S}$ is the impedance of the secondary circuit. When the secondary impedance is infinite the mutually induced emf appears as the voltage across the open ends of the secondary.

APPARATUS:

The apparatus provided for the experiment consists of the following.

1. A sine wave generator with output of about 10 V (rms) at 1000 Hz frequency
2. A digital multimeter (DMM) to be used as a voltmeter
3. A pair of coaxial coils wound one around the other on a cylindrical hollow piece of non magnetic non conducting bobbin
4. A piece of aluminum rod capable of fitting inside the bobbin
5. Two series of ten resistors mounted on an acrylic board: one consisting of 100 ohms resistors and the other of 10 ohm resistors with banana pin sockets
6. The required resistance for load $R_\mathrm{L}$ or sampling resistance $R'$ can be selected using this resistance board and connecting wire pieces. A separate acrylic board with resistance of 300 ohms is provided which can be used for sampling resistor when the other board is used for load resistance.
7. A set of five red and five black wires with banana pins at their ends (the pair of longer red and black wires is meant for use as DMM probes).

The output of the sine wave generator of frequency 1000 Hz is to be used as the alternating source voltage. Use the 20 V range of the DMM to measure rms ac voltages.

The magnitudes of currents $I_\mathrm{P}$ and $I_\mathrm{S}$ when needed are to be calculated from the measured voltages across known values of $R'$ in primary and $R_\mathrm{L}$ in secondary respectively.

EXPERIMENT

PART 1: Determining Resistance and Inductance of a Coil with Air-core and Aluminium Core (3,4 т.)

Connect coil 1 (with blue terminals) in series with a resistor $R'$ (to be selected from the resistance board) across the output terminals of the sine wave generator. The sine wave generator will be on before you start your experiment for stabilization of its output. Do not turn it off. Keep the output voltage amplitude maximum. (The DMM should show the output about 10 V).

The ac output of the generator may have some asymmetry. In that case, the readings of the DMM will show slightly different readings, when the input polarities of the DMM are interchanged between ‘V/Ω’ to ‘com’. To correct for the error due to asymmetry repeat each reading by interchanging the polarity of the probes of the DMM and take the mean of the two readings.

Choose the value of R´ to obtain $V_{\mathrm{R}'}$ and V approximately equal so that the systematic error in Z becomes negligible.

a) Measure the voltages $V_\mathrm{A}$, $V_{\mathrm{R}'}$ and $V$ as well as $V_\mathrm{O}$ across the terminals of the other coil. Determine the resistance $R_1$ and inductance $L_1$ of coil 1 (with blue terminals) and estimate the uncertainties in the values determined. [0,9 т.]

b) Connect the other coil (coil 2 with green terminals), make the necessary measurements and determine $R_2$ and $L_2$. Evaluate the uncertainties in the values obtained. [0,9 т.]

c) Now insert the piece of aluminium rod at the core of the coils and repeat the procedure to obtain the values of inductances $L_1$* and resistance $R_1$* and the uncertainties in them for coil1. [0,8 т.]

d) Make necessary measurements and determine inductance $L_2$* and resistances, $R_2$* of coil 2 when it has an aluminium core. Estimate the uncertainties in the values. [0,8 т.]

Note: Uncertainty calculations need not be done for parts 2, 3 and 4.

PART 2: Mutual Inductance and Coupling Constant (3 т.)

f) Mutual inductance $M$ can be obtained from the readings of $V_{\mathrm{R}'}$ and $V_\mathrm{O}$ (recorded in Part 1). Find the average values for air core as well as aluminium core coils. The relation between the mutual inductance and the self inductances of the coupled coils is given by $M = k\,(L_1L_2)^{1/2}$. Determine the value of $k$, the coupling coefficient in the two cases. [0,4 т.]

g) Select coil 1 (with blue terminals) as primary and coil 2 (with green terminals) as secondary. Connect the primary in series with the sampling resistor $R' = 300$ ohm across the output terminals of the generator. Connect across the secondary the variable load $R_\mathrm{L}$. The output voltage $V_\mathrm{O}$ is to be measured across $R_\mathrm{L}$. Change $R_\mathrm{L}$ and measure the voltages $V_\mathrm{A}$, $V_{\mathrm{R}'}$, $V$ and $V_\mathrm{O}$ corresponding to each value of $R_\mathrm{L}$. [0,8 т.]

h) A linear graph can be plotted combining various terms appearing in Equation (10) written in expanded form. Write the linear expression for plotting a graph whose slope can be used to obtain the value of $M$ and intercept to obtain the value of secondary reactance $X_\mathrm{S}$. [0,2 т.]

i) Calculate the necessary quantities using data from (g) to plot the graph corresponding to the expression obtained in the step (h) above. [0,9 т.]

j) Plot the graph and obtain the values of $M$ and $X_\mathrm{S}$. [0,7 т.]

Отговори

Решение

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Оригинал в Архива: exp2.pdf · официални решения: exp2_sol.pdf