University of Portland EE 271 Electrical Circuits Laboratory Experiment: Inductors I. Objective The objective of this experiment is to verify the relationship between voltage and current in an inductor, measure the charging time constant, and observe the response of a second order RLC system. II. List of Needed Components This experiment requires the following components: One 1 kω resistor Two 10 mh inductors One 0.1 µf capacitor III. Background The voltage across an inductor (see Figure 1) is proportional to the derivative of the current through the inductor: vv LL (tt) = LL dd dddd ii LL(tt). Figure 1: Inductor Voltage and Current University of Portland - p. 1 of 6 - Exp - Inductors.docx
If an inductor is initially uncharged, and it is charged by a D.C. source through a resistor, as shown in Figure 2, the current across the inductor is given by ii LL (tt) = VV ss RR 1 ee tt ττ where ττ = LL RR seconds. Figure 2: Circuit to Charge an Inductor IV. Prelab Assignment 1. Find vv LL (tt) in Figure 1 if ii LL (tt) = 0.01cccccc(2ππ5000tt). Also, find the peak value of vv LL (tt). 2. Find a formula for vv LL (tt) in terms of vv ii (tt) and vv RR (tt) in Figure 3. 3. Find a formula for ii LL (tt) in terms of vv RR (tt) in Figure 3. 4. Find the value of ii LL (tt) in Figure 2 after the inductor is charged for ττ seconds if Vs = 10V, R = 1 kω, and L = 10 mh. V. Procedure Part 1: Relationship between Inductor Voltage and Current The goal of this part of the experiment is to observe the relationship between the voltage across an inductor and the current through the inductor. There are two issues that complicate this process. First, the oscilloscope measures voltage, not current, so we will measure the voltage across a series resistor and calculate the current using Ohm s Law. Second, the ground leads on the oscilloscope probes are connected to earth ground through the oscilloscope, so if they are connected anywhere in the circuit besides ground, they will short that node to ground, which would change the circuit. So we cannot directly measure both the inductor and resistor voltage at the same time. However, we can measure both the source voltage and the resistor voltage at the same time, and then use the math feature of the oscilloscope to compute the voltage across the inductor (see Figure 3). University of Portland - p. 2 of 6 - Exp - Inductors.docx
Set the function generator to produce vv ii (tt), which is a 5 khz sinewave with 10 V peak and no D.C. offset. Build the circuit in Figure 3 with R = 1 kω and L = 10 mh. Figure 3: Circuit to Measure Inductor Voltage and Current Connect vv ii (tt) to Channel 1 of the oscilloscope and vv RR (ii) to Channel 2 (and both of the ground leads to ground). Use the MATH feature of the oscilloscope to compute and display vv LL (tt) using the formula you found in the prelab. If the display of vv LL (tt) is noisy, set the oscilloscope to take the average of several traces (press the Acquire Menu button, Mode, Average). Sketch vv LL (tt) and vv RR (tt) on the same axes and label the maximum and minimum voltages, and the period. Compute the inductor current ii LL (tt), and sketch vv LL (tt) and ii LL (tt) on the same axes and label the maximum and minimum voltages, and the period. Is the shape of vv LL (tt) and ii LL (tt) what you would expect? Briefly explain. Write a function to describe your measured ii LL (tt), and then compute the expected value of vv LL (tt) and compare it to the value that you measured by computing the percent error for the peak value. Change the function generator to a square wave, sketch vv LL (tt) and ii LL (tt) on the same axes, and label the period and maximum and minimum values. Is the shape of vv LL (tt) and ii LL (tt) what you would expect? Briefly explain. Change the function generator to a triangle wave, sketch vv LL (tt) and ii LL (tt) on the same axes, and label the period and maximum and minimum values. Is the shape of vv LL (tt) and ii LL (tt) what you would expect? Briefly explain. Turn the average mode off (press Acquire Menu button, Mode, Sample). Also turn off the MATH trace (press MATH button, Off). University of Portland - p. 3 of 6 - Exp - Inductors.docx
Part 2: Inductor Charging and Discharging The goal of this part of the experiment is to observe how the inductor charges and discharges, and to measure the time constant. Instead of using a switch to connect the inductor to either ground or Vs as shown in Figure 2, we will accomplish the same effect by using the function generator to produce a square wave voltage vv ii (tt) that alternates between 0 and Vs volts as shown in Figure 3. Set the function generator to produce a 5 khz square wave with a minimum value of 0V and a maximum value of Vs = 10V. It works best to first set the peak-to-peak voltage to 10V, and then adjust the offset so that the minimum value is 0V. Construct the circuit shown in Figure 3 above with R = 1 kω and L = 10 mh (same values as before). Sketch vv ii (tt) and ii LL (tt) on the same axes, and label the period and maximum and minimum values. We can measure the value of an inductor by measuring the time constant. Use vertical cursors to measure the time constant τ, which is the time required for the inductor current ii LL (tt) to charge to 63.2% of its final value. Place one of the cursors at the time when the inductor starts to charge, and place the other where the current rises to 63.2% of its final value. The time between the cursors is given by the time to the right of the Δ symbol. The voltage at the active cursor is given by the voltage after the @ symbol. Given your measured value of the time constant τ, compute the value of the inductor, and compute the percent error from its theoretical value of L = 10 mh. Replace the inductor in Figure 3 with two 10 mh inductors in series. Measure the time constant and compute the equivalent inductance. Compare the equivalent inductance with the theoretical value. Next replace the inductor in Figure 3 with two 10 mh inductors in parallel. Measure the time constant and compute the equivalent inductance. Compare the equivalent inductance with the theoretical value. University of Portland - p. 4 of 6 - Exp - Inductors.docx
Part 3: RLC Circuit In this part of the experiment, we will build a second order series RLC circuit and observe the capacitor voltage. Set the function generator to produce vv ii (tt), which is a 500 Hz square wave with a minimum value of 0V and a maximum value of Vs = 10V. Construct the circuit shown in Figure 4 with L = 10 mh and C = 0.1 µf. Connect vv ii (tt) to Channel 1 of the oscilloscope and connect vv cc (tt) to Channel 2. Figure 4: Series RLC Circuit Sketch vv ii (tt) and vv cc (tt) on the same axes, and label the maximum and minimum voltages, and the period of both the square wave and the oscillations. But wait, you may ask, if this is an RLC circuit, then where is the resistor? That s a good question that brings up the distinction between ideal circuit models and real components. A real inductor does not behave quite like the ideal inductor circuit model. A real inductor is constructed by forming a long wire into a coil. Since the wire has some resistance, a real inductor behaves more like an ideal inductor in series with a resistor (see Figure 5). Measure and record the resistance of the 10 mh inductor to verify that the inductor has resistance. Furthermore, the function generator does not behave quite like an ideal voltage source, instead it behaves more like an ideal voltage source in series with a 50 Ω resistor (see Figure 5). So, when we build the circuit in Figure 4 above using real equipment, it behaves more like the circuit model shown in Figure 5, which is an RLC circuit. So even though we did not explicitly add a resistor to our circuit, the circuit will nonetheless have resistance. The wires and breadboard also add to the total resistance, but their contributions are very small compared to that of the inductor and function generator. Measure the resistance of a wire to verify that its resistance is small compared to that of the inductor. University of Portland - p. 5 of 6 - Exp - Inductors.docx
(A real capacitor behaves more like an ideal capacitor in parallel with a resistor, but the value of the resistance is high enough that we can ignore it here.) Figure 5: Model for RLC Circuit with a Real Inductor and Function Generator Part 4: Resonant Frequency Using the circuit in Figure 4, set the function generator so that vv ii (tt) is a 1 Vpeak sinewave with no offset. Observe vv ii (tt) and vv cc (tt) while you change the frequency of vv ii (tt) from 1 khz to 100 khz. Find the resonant frequency of this circuit, which is the frequency where vv cc (tt) is the largest. With vv ii (tt) set to the resonant frequency, sketch vv ii (tt) and vv cc (tt) and label the maximum and minimum voltages and the period. Compare the resonant frequency to the theoretical value given by ff = 1 2ππ LLLL Hz. VI. Conclusion Write a paragraph that summarizes what you have learned in this lab. How is the voltage across an inductor related to the current through the inductor? When an inductor is charged through a series resistor, how long does it take for the current to reach 63.2% of the final value? How is the equivalent inductance computed for series and parallel inductors? Why does an RLC circuit oscillate? How is a real inductor different than the ideal inductor model? What is meant by resonant frequency of an RLC circuit? University of Portland - p. 6 of 6 - Exp - Inductors.docx