Apparent Power and Power Factor Lab

Objective:
Today's class meeting covered effective or RMS value. Through some calculation we found that the effective value is its root mean square. We can then use the rms values in terms of the average power where P = Vrms*Irms*cos(θ៴-θі). We also discussed about apparent power and power factor. The avg power mentioned ealier is the product of two terms where the product of Vrms and Irms is known as the apparent power and the factor cos(θ៴-θі) is known as the power factor. The power factor is the ratio of the average power to the apparent power. In other words, pf = P/S where S is the S = Vrms*Irms. The power factor in this case is dimensionless. Also, the power factor is the cosine of the phase difference between voltage and current.

Group Practice Problems:

1. The problem below proves the relationship between the effective and RMS values. They are in this case equal to each other as seen below. Therefore, we will not refer to the effective values but as RMS values.

Figure 1. RMS and effective values 

2. The problem below requires us to find the apparent power S and power factor pf which is dimensionless. We have a circuit where a resistor is connected in series with an inductor powered by an AC outlet of 210 and a frequency of 50Hz. The 210V is actually the Vrms. The first step is to convert the frequency to angular frequency and transform the circuit to its phasor domain. We will then find the Irms by using the equations I=Vrms/Z where Z is the equivalent and transformed t polar form for easier manipulation. The power factor is calculated by multiplying the Vrms and Irms. The power factor is the cos(θv-θi).

Figure 2. Calculating the power factor and the apparent factor

3. The problem below shows given voltage across  a load and a current through the element in the direction of the voltage drop. We are told to find the complex power, apparent power, the real power, reactive power, power factor and the load impedance. The results are shown below. 

Figure 3. Find the powers f a circuit given v(t) and i(t).

Lab Procedures and Results:

1. We are building a circuit . Given our theoretical values, we will calculate for the following: Irms, Vrms, apparent power, power factor, average power to load, average power dissipated by Rt, and the ratio between the average power Rt and the load. Our calculation can be seen below and we are required to find them while changing the RL= 10, 47 and 100Ω. 

Figure 4. Theoretical calculations procedures. 

2. Our theoretical calculation can be seen below for different RL in figure 5.

Figure 5. Theoretical and experimental values.

3. Values of Components: RT= 10.8ohms, L = 1mH, RL=10.8ohms ,RL=48.3 ohms, RL=99.1 ohms. The actual set up can be seen below. 

Figure 6. Actual set up of circuit.

4. Our experimental measurements are seen below in figure 7, 8, and 9 and can be seen in comparison with our theoretical values seen in figure 5.

Figure 7. Measurements based on RL = 10 ohms.

Figure 8. Measurements based on RL = 47 ohms.

Figure 9. Measurements based on RL=100 ohms.

5. The next step is to acquire the same data but introducing a 1 mF capacitor  parallel to the load. Measurement can be seen in figure 10, 11, and 12. The real value for the capacitor in the actual circuit is .949mF.

Figure 10. 1mF capacitor in parallel to the load where the RL = 10ohms.

Figure 11. 1mF capacitor in parallel to the load where the RL= 47ohms

Figure 12. 1mF capacitor in parallel to the load where the RL= 100ohms.

Lab Learning Outcome:

The objective of the lab is to compare our theoretical results with our experimental results for Irms and Vrms for different load resistor of  RL=10.8ohms±.1 ,RL=48.3 ohms±.1, and RL=99.1 ohms±.1. The load is composed of the inductor which is L = 1mH in series with the RL. There is also and Rt in series with the load which is represented as the transmission line which has a resistance of RT= 10.8ohms±.05. We calculated our theoretical Irms and Vrms based on our formulas seen in figure 4 where are results are seen in figure 5. When comparing them to our experimental results, we acquired a percent error for the Irms of 3.16% when RL=10ohms, 2.39% when RL=47ohms, and 1.46% when RL = 100ohms. The Vrms percent error are .16% for RL=10ohms, 2.47% for RL=47ohms, and 2.05% for RL=100ohms. By having a small percent error, we can conclude that our theoretical calculation serves its purpose in solving for the Irms and Vrms, as well as calculate the apparent power, power factor, and the phase shift.