ii) Resistor and Inductor in a series circuit

In this simple series circuit (Fig. 1), the AC current is the same through both the resistor and the inductor. It can be shown, by using Ohm’s law, that:

(6)

Thus the voltage across the inductor increases while the voltage across the resistor decreases when the frequency is increased, so that U ₀remains constant:

(7)

(8)

where U ₀is the amplitude of the voltage across both the resistor and inductor, and ϕ is the phase difference between the voltage across the resistor and the voltage across both the resistor and inductor. Thus the phase difference between the inductor voltage and current remains constant, whereas the phase difference between current and the voltage across both the resistor and inductor increases with frequency.

This theoretical background is sufficient in order to perform this experiment. However, if you require additional details or clarifications, or would like to know more about this topic, then you may refer to Adams and Allday ̓s Advanced Physics page 222 or Serway ̓s Essentials of College Physics pages 551-555.

Experimental Procedure

In order to obtain good quality data the Oscilloscope traces must be well focused and the intensity set low. Remember that you can use the Position knobs to move the traces on the Oscilloscope screen and place them conveniently with respect to the vertical or horizontal axes.

Your report should include the following sections: Introduction (no more than 5 sentences outlining the main objectives, equipment and expectations), Experimental Procedure (only the requirements stated in section 3), and Conclusion (brief summary, ~ 5 sentences, of the purpose of this experiment, your results and the meaning/interpretation of your results). It is not required to describe the experimental procedure steps in your report.

Part A.

1) Check that the inductor and resistor have been correctly connected in series with the signal generator, as shown in Fig 1. below;

Black CH 1, 2

Oscilloscope

Oscilloscope

Figure 1. Schematic diagram of the series RL AC circuit

Function Generator

2) Check that Channels 1 and 2 of the Oscilloscope have been connected to the circuit, as shown in Fig. 1 above;

3) Set the Oscilloscope Vertical Mode to Dual in order to visualize both traces at the same time. Using the Function Generator vary the AC frequency between 10 and 220 kHz. State briefly (1 sentence) what you observe on the oscilloscope screen. Based on your observations, what voltage is displayed by Channel 1, and what voltage is observed in Channel 2?

4) Choose 10 AC frequencies between 10 and 220 kHz. Find V _{0 R} and V _{0 L} at each frequency;

5) Record your results in a table with the following headings: f, V _{0 R} , U ₀, V _{0 L} , and . Explain briefly how you found these quantities. Write down the equations that you used, and show sample calculations where applicable. It is sufficient to provide sample calculations for only one of the ten frequencies;

6) Plot a graph of against frequency. Is your graph consistent with the theoretical predictions? Why or why not?

7) Find the gradients of the best-fit (trend) line, the line of maximum gradient and the line of minimum gradient using the box (rectangle) method. The best-fit line and its gradient can be obtained only by using the “Add Trendline“ feature in Excel. No other method will be accepted. A complete graph includes these three lines, the box, and the best-fit line equation displayed on the graph;

8) Calculate the inductance L and its uncertainty ∆ L based on the gradients obtained from your graph.

Part B.

1) Using the same circuit and Oscilloscope connections, measure the period T of any of the Oscilloscope traces, for each of the following three frequencies: 100 kHz, 150 kHz and 200 kHz. If the inverse of T does not equal the Function Generator frequency, then your Oscilloscope is not calibrated properly. In that case you should check and make sure that the SWP VAR knob is rotated all the way to the right;

2) Then, switch the Vertical mode to Dual and measure the separation ΔT of the two Oscilloscope traces at each of the three frequencies mentioned above. It is recommended that you choose adjacent time-axis intersection points in order to measure ΔT, as shown in Fig. 2 below. The phase difference is then:

(9)

3) Calculate ϕ using equations 8 and 9, for each of the three frequencies mentioned above;

4) Record in a new table: T, ΔT, ϕ (in degrees) calculated with equation 9, ϕ (in degrees) calculated with equation 8, and the difference between the two ϕ values for each of the three frequencies mentioned above. Note: in equation 8 you must use the inductance value found in Part A;

5) Are your newly tabulated results consistent with the theoretical predictions? Why or why not?

Figure 2. Dual trace mode view of the two Oscilloscope traces showing their period T and time-axis separation ΔT.

Part C.

1) Now switch the Oscilloscope’s Time Base (time-axis control knob) to the XY mode. This will allow you to visualize the Lissajous figure corresponding to the two Oscilloscope traces. This Lissajous figure is obtained by plotting the voltage displayed by one Oscilloscope channel against the voltage recorded by the other Oscilloscope channel. You can find which voltages are plotted on the vertical and horizontal axes by switching the Time Base back and forth;

2) Use the Position knobs to move the Lissajous figure on the Oscilloscope screen until it is symmetric with respect to the origin of the axes. Then, sketch the Lissajous figure for each of the three frequencies mentioned in Part B, and indicate which voltage is plotted on each axis. Also, you need to determine the coordinates of the points of intersection with the axes (4 points), and the coordinates of the maximum and minimum voltages on each axis (4 points). You are required to sketch these curves like regular graphs, which have adequate titles and axes labels.

Part D. List the main sources of errors in this experiment.

Conclusion

Summarize briefly (~ 5 sentences) the purpose of this experiment, the equipment used, your results and the meaning of your findings. These should not be restricted to the main objectives outlined in the Introduction.

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