4
CPS NU Physics Laboratory
PLEASE DO NOT WRITE ON THIS PAPER AND RETURN IT AT THE END OF THE SESSION
Inductance in an AC Circuit
(See Adams and Allday P 222 and 223)
Introduction
In this experiment you will use an oscilloscope to investigate and measure the voltage across a resistor, and the voltage across an inductor, in a simple series AC circuit. These voltages are to be measured as a function of the frequency of the applied voltage from the signal generator. In addition you will observe the phase difference between the voltage across the inductor and the current passing through it.
Vocabulary
Inductor, Inductance, reactance, frequency, signal generator, coil, sinusoidal, Oscillator, Oscilloscope
Theory
i) Reactance and Resistance
A resistor provides resistance to the flow of current in a circuit and the relationship between voltage across the resistor and the current is given by Ohm’s Law. The resistor converts electrical energy into heat energy – it is said to dissipate electrical energy. These basic principles apply whether the current in the circuit is constant (DC) or whether it is a (sinusoidal) alternating current (AC). Thus if the voltage across the resistor is given by:
Where ω is the angular frequency,
then Ohm's Law remains true, and the current and voltage oscillate in phase and with the same frequency. The current is given by:
Where
Now consider a coil of wire (called an inductor) in a circuit and suppose it has a negligible resistance. When the current through it changes with time, an electromotive force (emf) is set up in the coils of the wire to oppose the change. This follows from a fundamental law of electromagnetism called Faraday’s Law. (An emf produces energy to drive electrons around a circuit – just like the voltage of a battery).
The faster the change in the current, the larger the emf opposing it. This means that when an AC voltage is applied to a coil an emf is set up to oppose the applied AC voltage. We can say that the coil produces a “resistance” to the alternating current. Also the higher the frequency of the applied voltage, the larger this “resistance” becomes. This “resistance” is called a reactance and is quite different from the normal resistance of a resistor. The reactance does not dissipate electrical energy as heat. Also the reactance is not independent of frequency but is directly proportional to it.
It can be shown that the reactance of an inductor is given by:
Where ω, equal to 2πf, is the angular frequency of the current and L is a quantity called the inductance. This inductance is a constant for a given coil and depends on the number of turns of wire as well as the core material. It is clear that the reactance for a given inductor is proportional to the frequency of the AC.
Just as resistance is the ratio of voltage to current for a resistor, so the reactance is the ratio of the amplitude of the voltage to the amplitude of the current for an inductor.
It can also be shown that the voltage across the coil is 900 out of phase with the current through it. (The current leads the voltage by 900).
You can think of the reactance as similar to inertia in mechanics. A change of velocity of a moving object is opposed by the inertial mass of the object, and the faster the change the more the inertial mass opposes the change. In the same way the reactance of an inductor opposes the change in current flowing through it, and the faster the change the larger the “opposition” to the change.
ii) Resistor and Inductor in a series circuit
In this simple series circuit, the AC current is the same through both the resistor and the inductor. The amplitude of the voltage across the resistor is given by:
The amplitude of the voltage across the inductor is given by:
Therefore the ratio of these two voltages is given by:
This result is exactly what we should expect. The voltage across the inductor will get much larger in comparison with that across the resistor since the reactance increases with frequency but the resistance does not change.
Experimental Procedure
Part 1.
1) Connect the inductance and the resistor in series with the signal generator as in Fig 2.
Figure 2. Schematic diagram of the circuit.
2) Connect channel 1 on the oscilloscope as shown in the circuit to measure V0R. Inductor is connected to channel 2 to measure V0L.
3) Trigger the oscilloscope on channels 1 and 2 to observe the wave forms of the voltages V0R and V0L.
4) Use the displays on the oscilloscope to measure the amplitudes of V0R and V0L for 8 frequencies (f) ranging from 10 kHz to 220 kHz.
5) Present your results in a table with the following headings:
Frequency f, V0R, V0L,
6) Plot a graph of against the frequency.
7) Find the gradients of the line of best fit, the line of maximum gradient and the line of minimum gradient. Calculate the value of the inductance L and estimate the error in your L value.
Note: You will need to take care if your data are to be of good quality. Make sure that you have the oscilloscope trace well focused and the intensity set low. Remember you can use the X and Y shift controls to move the traces about on the screen and place them conveniently with respect to the screen measurement scale.
Part 2.
1) Use the same circuit and oscilloscope connections. For each of the following three frequencies: 100 kHz, 150 kHz and 200 kHz, estimate the phase shift between the voltage across the inductance and the resistor.
They will appear as below due to the phase shift.
2) Measure the period (T) of one oscillation of either V0R or V0L (they are the same). Then measure the distance, ΔT, between two adjacent peaks of V0R and V0L. The phase shift is then:
Φ = radians
3) Present data in a table. For each of your three frequencies state the values you obtain for ΔT, T, and the phase shift.
4) Compare your values with the expected phase shift of 90°.
Now switch the time base to XY display. The spot on the oscilloscope is being driven in the x-direction by the oscillating voltage across the resistance and in the y-direction by voltage across the inductance which is 90° out of phase with that across the resistance.
5) Measure the voltage on the X axis (across the resistance) and the voltage on the Y axis (across the inductance) for 100 kHz, 150 kHz and 200 kHz. Sketch the figures for each frequency indicating the voltages on the Y axis and on the X axis.
Experiment Questions:
1) State all sources of errors in this experiment.
2) Explain how you can increase the inductance of a coil.