28-May-2015: Signals with Multiple Frequency Components

26-May-2015: Apparent Power and Power Factor

PURPOSE

The purpose of this lab was to implement what we learned about apparent power and power factors in a real-life situation.

PRE-LAB

Figure 1

Before setting up our experiment, we did some initial calculations. We first found the Thevenin impedance, which allowed us to find the RMS current (Irms) and voltage (Vrms) across the load. In turn, we were able to calculate the average (Pavg) and apparent power (S) from these values. Finally, we took the ratio of these two variables to find the power factor (Pf). However, after looking over the values, particularly Zth, we realized that we did the calculations incorrectly and were unable to compare them to the experimental values.

PROCEDURES

Figure 2

After completing our initial calculations, we set up the circuit as shown in Figure 2. We used channel 1 of our oscilloscope to measure the input voltage and channel 2 to measure the voltage across the load. Then, we set up a math channel to find the current through the load. Then, we had the oscilloscope display the RMS values of these measurements.

Figure 3: RT = 10 Ω

This was the resulting graph when RT = 10 Ω. As it can be seen from the image, Vrms and Irms were 599.6 mV and 68.12 mA, respectively. From these values, we were able to find the apparent power delivered to the load (S = VrmsIrms), which was equal to 40.84 mW. When compared to the theoretical value of 11.86 mW, they were nowhere near each other. As mentioned before, we could only assume that we made an error in our pre-lab calculations. In addition, we found the average power delivered to the load with the equation PL = VrmsIrmscos(θv - θi). However, in order to do so, we had to first find the phase angle between Vrms and Irms with the equation θv - θi = Δt/T * 360°. By utilizing the graph, we were able to find the phase angle to be 17.06°. This gave us an average power of 39.04 mW, which gave us a power factor of 0.956.

Figure 4: RT = 47 Ω

When RT = 47 Ω, Vrms was 129.2 mV and Irms was 68.11 mA. This gave us an apparent power of 8.800 mW. Since there wasn't a phase shift between the voltage and the current, the average power delivered to the load was the same as the apparent power. Therefore, the power factor was equal to 1.

Figure 5: RT = 100 Ω

Next, we replaced RT with a 100 Ω resistor. The resulting graph is shown in Figure 5. From the graph, we were able to find that Vrms and Irms were 201.4 mV and 68.27 mA, respectively. From these values, we found S to be 13.75 mW. Then, we found the phase angle to be 54.18°. This gave us a Pavg of 8.047 mW and a Pf of 0.585.

14-May-2015: Inverting Voltage Amplifier

PURPOSE

The purpose of this experiment was to implement an inverting voltage amplifier to an AC voltage.

PRE-LAB

Figure 1

Figure 2

Figure 3

We began this experiment by analyzing the circuit diagram shown above (Figure 1). We first converted the frequencies that we were going to use in the experiment into angular frequencies as shown in Figure 2. Then, we found the theoretical gains and phase shifts of the circuit with the equation displayed in Figure 3.

PROCEDURES

Figure 4

After completing our pre-lab exercises, we set up the circuit as shown above in Figure 4. We then applied a voltage with an amplitude of 2 V and a frequency of 100 Hz to the circuit. The resulting output voltage is shown in Figures 5 and 6. We then applied two more voltages with the same amplitude but with frequencies of 1 kHz, and 5 kHz, respectively. The resulting output voltages from these inputs are displayed in Figures 7 through 10.

Figure 5: Frequency at 100 Hz (input)

Figure 6: Frequency at 100 Hz (output)

Figure 7: Frequency at 1 kHz (input)

Figure 8: Frequency at 1 kHz (output)

Figure 9: Frequency at 5 kHz (input)

Figure 10: Frequency at 5 kHz (output)

Based on the graphs above, we were able to find the phase shift angles at each frequency. We were able to do this by dividing the time difference between the peaks of the input and output voltages by the period of the graphs and multiplying by 360 degrees (θ = Δt/T * 360°). We also found the gains by dividing the amplitude of the output voltage by the amplitude of the input voltage. These calculated values are shown in Figure 11.

Figure 11

Finally, we compared the theoretical values that we found in the pre-lab and compared them to the values that we found experimentally. The percent difference between these values are shown above in Figure 11. It can be seen from percent differences that our experimental values were pretty accurate.

12-May-2015: Phasors - Passive RL Circuit Response

PURPOSE

The purpose of this lab was to find the gains and phase differences between the input current and input voltage at three different frequencies and compare them to the theoretical values.

PRE-LAB

As a pre-lab exercise, we found the cut-off frequency (ωc) based on the circuit elements R and L. Then, we calculated three frequencies that were proportional to this cut-off frequency as shown in Figure 1 (we also converted these frequencies from rad/s to Hz). Using these values, we were able to derive the gains and the phase differences of the circuit at each frequency (Figure 1).

PROCEDURES

Figure 2

After completing the pre-lab, we set up the circuit as shown in Figure 2. We applied a 1 V sinusoidal voltage to the circuit and measured the output voltage across the resistor (vr(t)) and across the inductor (vl(t)). We also set up a math channel to find the input current by using the formula i(t) = [v(t) - vl(t)]/R. The resulting graphs at each frequency are shown below in Figures 3, 4, and 5.

Figure 3: 748 Hz

Figure 4: 74.8 kHz

Figure 5: 7.48 kHz

From the resulting graphs, we were able to find the gains and phase differences between i(t) and v(t). We found the gain by dividing the amplitude of i(t) by the amplitude of v(t). We found these values to be 0.0143, 0.0021, and 0.0197, respectively. These were very close to the values found in the pre-lab. As a matter of fact, the percent differences between the theoretical and experimental values were only 4.67, 0.51, and 7.08 percent, respectively. Therefore, we concluded that our experimental values were valid. In addition, the phase differences were calculated by using the formula ϕ = Δt/T x 360°. The resulting values were 45.78, 88.86, and 5.39 degrees. These values were also pretty accurate as their percent differences were 2.14, 5.47, and 5.07 percent, respectively.

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.

30-Apr-2015: RLC Circuit Response

PURPOSE

The purpose of this experiment was to observe the response of an RLC circuit and analyze the resulting graphs to find missing variables.

PRE-LAB

Figure 1

To begin our lab, we sought to find the damping ratio (ζ) and the natural frequency (ωo) based on the given circuit elements (Figure 1). However, in order to do so, we had to first measure the actual resistance values of the resistors. These measured values are included in the schematics of the circuit in Figure 1. From these values and the other circuit elements, we found the neper frequency (α) and the natural frequency (ωo). Finally, we found the ratio of these two values to find the damping ratio (ζ)

PROCEDURES

Figure 2

After the pre-lab, we constructed the circuit as shown in Figure 2. We had the analog discovery connected to the second resistor (R2) to measure the voltage across it, which we considered to be Vout. Then, we applied a 2 V step input to the circuit at varying frequencies until the circuit was able to reach steady-state in between pulses.

Figure 3

Figure 4: Close up of Figure 3

The resulting graphs are shown above in Figures 3 and 4. From Figure 4, it can be seen that the the overshoot (Mp) was approximately 35 percent. This is because the initial voltage was around 40 mV and the voltage decayed past 0 mV by 14 mV (Mp = 14/40 * 100% = 35%)

28-Apr-2015: Series RLC Circuit Step Response

PURPOSE

The purpose of this lab was to implement what we learned about an RLC circuit in series in a real-life experiment.

PRE-LAB

Before setting up our circuit, we set up a second order differential equation to relate Vout and Vin. We also did some calculations to estimate the damping ratio (ζ), the natural frequency (ωo), and the damped natural frequency (ωd). In order to do so, we had to first find the actual values of the circuit elements. We found the resistances of the resistor and the inductor to be 1.4 Ω and 1.7 Ω, respectively. Furthermore, we found the inductance of the inductor to be 0.999 mH and the capacitance of the capacitor to be 0.437 µF. From these values, we found the necessary variables as shown in Figure 1.

PROCEDURES

Figure 2

To begin our experiment, we constructed the RLC circuit as shown in Figure 2. It consisted of a resistor, an inductor, and a capacitor, all connected in series. Then, we applied a 2 V step input at 1 Hz to the circuit and plotted Vout and Vin on the oscilloscope. The resulting graphs are shown in Figures 3 and 4.

Figure 3: Under-damped circuit (1)

Figure 4: Under-damped circuit (2)

For an under-damped circuit, the response oscillates at the damped natural frequency. In order to find the damped natural frequency, we multiplied the inverse of the period (which we found by analyzing the graphs in Figures 3 and 4) by 2π. We found this value to be 52359.9 rad/s. This experimental value was somewhat close to the theoretical value of 47835.3 rad/s. In fact, the percent difference between the two values was 9.459 percent. Although this number was slightly bigger than what we would have preferred, we believed that the results were acceptable since there were many possible sources of error in this experiment such as the actual capacitance of the capacitor.

Figure 5

For the second part of the lab, we estimated the resistor value that would cause the circuit to be critically damped (Figure 5). As it can be seen from the image, we found this value to be 93.93 Ω. However, since we did not have this resistor value available for use, we chose a 100 Ω resistor instead. The measured value of the resistor (97.8 Ω) is shown in the schematic of the revised circuit in Figure 5.

Figure 6: Critically damped circuit

Figure 6 shows the response of the revised circuit to a 2 V step input oscillating at 1 Hz.