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Introduction

The purpose of this experiment was to learn about the characteristics of inductors and capacitors. In the pre-lab calculations the students gained experience solving first order differential equations. During the experiment, functions were plotted on the oscilloscope and in PSPICE to demonstrate the voltage versus time relationship when charging and discharging inductors and capacitors.

Procedure

Preliminary calculations:

1. Figure 7.3 shows how a voltage source can be modeled as an ideal voltage source with a resistance, Rint. Rint can be found by placing a known resistance R across the open terminals. This allows the internal resistance to measured and be used in circuits that are energized by voltage sources.

Voc = Vab, when a-b is open circuited, and VR = Vab, when a resistance R is connected between a-b. Rint is found by the following equation: Rint = [R(VOC-VR)]/VR.

2. For figure 7.4, it was calculated that Vc(t) = VP + VH = Vs(1-e-(t/RC)) when the switch is open.

It was also found that Vc(t) = VP + VH = Vse-(t/RC), when the switch is closed.

3. An inductor can be modeled as a resistor in series with an ideal inductor. This is called an “actual” inductor. For Figure 7.5, Rdc = [R/(1-VL)]VL.

Main Procedure:

Step 1:

The equation Rint = [R(VOC-VR)]/VR from part one of the pre lab calculations was used to find Rint for the square wave generator, with the generator set to 2kHz. Values of R were used that gave VR readings in the range of 1/3VOC to 2/3VOC. It was found that Rint = 84.3Ω. This value will be included in the total resistance in the remainder of this lab.

In the next steps, it will be shown how RC, RL, and RLC circuits can be demonstrated with the use of a square wave generator. Considering Figure 7.7, if the switch is left in position 1 for a long enough time, the capacitor charges to V volts. At t = 0, when the switch position is changed to postion 2, the capacitor discharges and it gives the natural response of the RC circuit. Figure 7.8 shows the same result repeatedly.

Step 2:

The circuit in Figure 7.8 was constructed using R = 470Ω and C = 0.1µF. The function generator was set to give a 1V peak-to-peak square sine of 2 kHz. Using the oscilloscope to observe VC(t), a portion of the waveform was sketched.

The theoretical time constant of the circuit is t = (0.1x10-6F)(470Ω) = 47µs. This matches with the time constant given by the graph. The sketch was used to find the time constant by seeing where the voltage is 0.367Vs.

Step 3:

Next, part 2 was repeated for R = 6.8kΩ and C = 0.1µF.

The theoretical time constant of t = 680μs matches with the time constant obtained from the graph.

Step 4:

Using the procedure shown in part 3 of the pre-lab, Rdc for a 68mH inductor was found to be

Rdc = 84.3Ω.

Step 5

The circuit shown in Figure 7.9 was constructed using R = 680Ω and L = 68mH. With the amplitude of Vs(t) set to a 1V peak-to-peak square wave, the voltage was sketched and the time constant was determined.

The theoretical time value was calculated that t = L/R = 68mH/(680Ω-84.3Ω) = 0.08897ms. This matches with the time constant obtained from the graph.

Step 6:

Using the procedure above, the class was to find the dc resistance and inductance of an unknown inductor supplied by the lab instructor. The equation used was t = L/R. For this group, test inductor #8 was used. By plugging in the time constant t = 7.2ms and R = 238Ω + 680Ω, it was found that Lof #8 = 6.6096H. This was 12% away from the actual value of 5.9H, which was revealed by the professor after our calculations were complete.

Step 7:

The circuit in Figure 7.7 was constructed in PSPICE and vc(t) for t>0 was plotted.

Next, the same thing was done for C = 0.1μF and R = 470Ω and R = 6.8Ω.

These PSPICE plots match with the earlier plots from the oscilloscope.

Step 8:

The group then used PSPICE to obtain resistor voltages that were obtained in parts 5 and 6. These match the experimental results. The first circuit was designed with R = 84.3Ω.

Measuring 0.633V vertical on the plot, gave a time constant of t = 0.08943ms. This matches with the value t = 7.2ms from step 5. The next circuit was designed with R = 238Ω.

This plot, gave a time constant of t = 7.082ms. This is close to the value t = 7.2ms from step 6.

Discussion

In step 6, the found value of test inductor #8 had an error of 12%. This is relatively far from the true value but, it was as close to the true value as the method and equipment would allow. It is also important to note that the uncertainty of the capacitor used in class was ±20%. This value was found by observing the “M” on the capacitor. Capacitors are marked differently than resistors and inductors.

During the experiment, it was helpful to use online charts to understand the capacitor value and temperature tolerances. The lab temperature was well within the normal range for this experiment, but there are real world applications where circuits must be able to function in extreme high and low temperatures. In future experiments it would be good to explore how temperature affects capacitor values.

Conclusion

The voltage versus time plots above show that capacitors and inductors charge and discharge within just a few milliseconds or just a couple hundred microseconds. Although they never reach a value of zero, the steepest voltage decline happens during this short time period. Voltage versus time for these types of circuits is represented by differential equations. Therefore it is vital to continue practicing obtaining and solving differential equations for first order circuits.

1Vdc

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470

1.000V

1Vdc

.1u

470

1.000V

1Vdc

.1u

6.8k

1.000V

1Vdc

.1u

6.8k

1.000V

680

68mH

1Vdc

1.000V

84.3

679.5uV

763.7uV

680

68mH

1Vdc

1.000V

84.3

679.5uV763.7uV

680

6.61H

1Vdc

238

680

6.61H

1Vdc

238