7-May-2015: Impedance

PURPOSE

The purpose of this lab was to use Ohm's law to find the impedances of different circuit elements experimentally and compare them to the theoretical values.

PRE-LAB

Figure 1

For our pre-lab, we set up equations to find the impedances of the three circuits shown above (Figure 1). The impedance of the resistor circuit was easy to find as it was simply equal to the sum of the resistance values. On the other hand, the impedances of the two other circuits depended on the frequency of the input voltage.

PROCEDURES

Figure 2

After doing our initial calculations, we set up the circuit shown above. We then applied 2 V to the circuit at frequencies of 1 kHz, 5 kHz, and 10 kHz. We measured the voltage across the fixed resistor (vr(t)) and the voltage across the "unknown" resistor (v(t)). In addition, we set up a math channel to plot the current running through the circuit by dividing the resistor voltage by the resistor's resistance value. The resulting graphs are shown below in Figures 3, 4, and 5.

Figure 3: 1 kHz

Figure 4: 5 kHz

Figure 5: 10 kHz

From these graphs, we were able to find the impedance of R by dividing v(t) by i(t). We found this value to be 101 Ω at all three frequencies. Since the measured value was 100.3 Ω (Figure 1), the percent difference between the theoretical and experimental values was only 0.7 percent.

Next, we replaced the 100.3 Ω resistor with a 1 µH inductor (Figure 6). We repeated the same process as before and applied a voltage at the different frequencies. Once again, we measured vr(t) and v(t) and set up a math channel to find i(t). The resulting graphs are shown below (Figures 7 through 9).

Figure 7: 1 kHz

Figure 8: 5 kHz

Figure 9: 10 kHz

From these graphs, we were able to find the impedance of the inductor utilizing the same technique as earlier (Z = v(t)/i(t)). However, this time, the results were not as accurate. At 1 kHz, we found the impedance to be 171 Ω, which was 89 percent off from the expected value of 1591.5 Ω. At 5 kHz. the measured impedance was 63.3 Ω, while the theoretical value was 318.3 Ω. The percent difference between these values was 80 percent. Finally, at 10 kHz, the experimental value was 53.8 Ω and the expected value was 159.2 Ω, which were 66 percent off.

Figure 10

For the third part of the experiment, we replaced the inductor with a 100 nF capacitor. We repeated the same steps as the first two parts of the experiment. The graphs of vr(t), v(t), and i(t) are displayed below in Figures 11 through 13.

Figure 11: 1 kHz

Figure 12: 5 kHz

Figure 13: 10 kHz

After analyzing these graphs, we found the impedances to be 6.37 Ω, 32.8 Ω, and 65.0 Ω, respectively. These were very close to the theoretical values of 6.3 Ω, 31.4 Ω, and 62.8 Ω. In fact, the percent error between the values were only 1, 4, and 3 percent, respectively.