Next, we reviewed apparent power, average power and the power factor. Apparent power, denoted as S, is the product of the rms values of the voltage and current, or 1/2 times the maximum voltage and current. This is so because the rms values of voltage and current are max voltage/current divided by the square root of 2. Moving on, the average power is the apparent power times the power factor, or the cosine of the difference between the phase angle of voltage and current. The power factor is also the cosine of the angle of the load impedance. We then did a problem involving these different powers.
After, we learned about complex power, which is given by multiplying half of the voltage by the complex conjugate of the current. It is also given by multiplying the apparent power by the sine of the phase difference between voltage and current. The apparent power is given by the average power plus j times the complex power.
We then learned about the power triangle, which provides the apparent power (hypotenuse), the average power (leg on x-axis), complex power (leg parallel to y-axis) and the power angle. When the triangle is in the 4th quadrant, the load is capacitive and therefore the pf is leading. When the triangle is in the 1st quadrant, the load is inductive and therefore the pf is lagging. We then did a problem involving the power triangle.
Next, we learned about power factor correction in that the power factor can be increased simply by installing a capacitor in parallel to an inductive load, since purely inductive loads operate at low pf. A problem was then solved involving power factor correction. We then did a lab titled "Apparent Power and Power Factor Correction".
LECTURE:
Above is the derivation for the rms value for current. All other rms values follow the same equation, with the fact that a value other than current is replaced with the current in this equation.
In this problem, the objective was to determine the rms value for this sinusoidal wave. The formula for rms was used in this problem, with the value in the formula being the equation of this time-varying signal, or 2Vp*sin^2(wt). The rms value was found to be the square root of 1.5 times Vp.
In the above picture, we came up with an acronym to remember the phase angles of voltage and current for inductors and capacitors. This acronym is Eli the Iceman. Eli tells use that the inductor's voltage (emf) leads the current by pi/2. Ice tells us that the voltage (emf) of the capacitor lags the current by pi/2.
In this example, the apparent and average power of the load (30 ohm resistor and 0.5 H inductor) were determined, along with the power factor.
In this example, using the provided voltages and currents the apparent power, average power, complex power, power factor and impedance was determined. These values were then plotted on a power triangle. Because the power triangle is in quadrant iv instead of i, it indicates that the load is capacitive (or a leading pf).
In this problem, the necessary capacitance was determined in order to bring the power factor from 0.8 up to 0.95. First, the difference in complex power before and after the power factor correction was determined, then it was divided by the frequency and the rms voltage squared to determine to need a capacitance of 310 uF to bring up the power factor from 0.8 to 0.95.
LAB:
Apparent Power and Power Factor Correction:
Purpose:
The purpose of this experiment was to analyze apparent power and power factor of a load and to determine the effects of changing the resistance of the load on the apparent power, average power and power factor. The load is connected to a resistive network. Expectations were determined and those were tested on a real circuit and via measurements. It was expected that a larger difference between apparent and average power correlated to a small power factor and that the power provided to the load would be much less.
Prelab:
In the prelab, different values were determined at different resistances for the load. The inductive reactance, however, stayed the same, along with the transmission resistance. In the prelab, the RMS current delivered by the source, the RMS load voltage, the average power delivered to the load, the apparent power delivered to the load, the load's power factor, the average power dissipated in the transmission resistor and the ratio between the average power dissipated in the transmission resistor and the average power delivered to the load were determined.
Procedure/Data:
First, the circuit seen in the prelab was first constructed, as seen above. In this picture, the 10 ohm resistor is placed in the load impedance. A wavegen channel was used to aply the input voltage. Channel 1 was used to measure the input voltage and channel 2 was used to measure the voltage across the load. The oscilloscope window was then used to determine the input voltage and voltage across the load. A math function ws also created to determine the current through the circuit. This oscilooscope window for the 10 ohm load resistor is shown below.
The same was then done with a 47 ohm load resistor instead of a 10 ohm load resistor. The reactance of the inductor stayed constant throughout the whole experiment.Below is the circuit with the 47 ohm load resistor.
Again, Ch1 was used to measure input voltage and Ch2 was used to measure the load voltage. The oscilloscope window is shown below along with a math channel to read current.
Again, the same setup and procedure was done again, with the exception that now a 100 ohm load resistor was used in replacement with the 47 ohm load resistor. The circuit is shown below.
Below is the oscilloscope window for this circuit. The same channels were used as prior.
The oscilloscope windows were then used to determine the data obtained and needed from this experiment. This data is the same as that calculated in the prelab. This data is shown in the data analysis section below.
Data Analysis / Conclusion:
Above is the data obtained from the oscilloscope windows measured in the procedure and displayed in the data section. What's inside the table is the expected results, and what's outside is the obtained data from the experiment. Looking at the 10 ohm load resistor, the experimental load voltage (840 mV) is larger than the theoretical (626 mV), with a percent difference of 34.2%. It is unknown exactly why this might have occurred, but it is possible that because small impedances were used, any small change in the resistance or reactance resulted in a larger change in voltage and current. Agsain, the current is the same; a percent difference of 36.8% between the expected (19 mA) and the experimental (26 mA). This also applies to the other trials involving the 47 ohm and 100 ohm load resistors, which helps verify that it is most likely due to the inductor not being 1 mH but actually deviating from it. Since the inductance is very small, the change affected the results dramatically.
Below is the experimental power factor, apparent power, average power, power dissipated by the transmission resistor, and the ratio between the transmission power and load power.
As can be seen, with an increasing load resistor, the power factor increases along with the ratio between transmission power and load power. However, the apparent power decreases with increasing load resistance. An interesting note is that the average power for both the load and transmission is largest when the load resistance is at 47 ohms.
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