Phasors: Passive RL Circuit Response Lab

Objective:
We entered the world of alternating voltage where we have sinusoids to deal with. For instance, if we have a v(t)=Vsinwt, the v is the amplitude (vertical stretch), w is the angular frequency in rad/sec(stretches the wave), and can have a phase shift or vertical shift.  We can also see if we graph two sinusoidal voltages where one has a phase shift, we can see which voltage leads or lags the other. When we deal with sinusoids and adding them, we can use a graphical technique which makes calculations easier. the horizontal axis represent the magnitude of cosine and vertical axis represent the magnitude of the sine. Steinmetz is the father of AC analysis and now sinusoids can be expressed in terms of phasors. Phasors is a complex number that represents the amplitude and phase of a sinusoid. Our complex number can be written or represented in three ways: rectangular form, polar form, and exponential form.

Group Practice:
1. The image below show how to add two sinusoidal voltages by using a a graphical technique. the length is calculated by calculating the magnitude and finding the angle using trigonometry.

Figure 1. Calculating the magnitude and direction of the sinusoidal voltages.

2. The problem below shows two sinusoidal voltages and are told to find the phase between the two voltages. Using the graphical technique, we can see that the phase is 90-50-10 = 30 degrees.

Figure 2. Calculating the phase between two sinusoidal voltages using the graphical technique.

3. We are given a complex number in rectangular form We are told to rewire it in polar form, therefore, we must find the magnitude, as well as the phase angle using inverse tangent.

Figure 3. Rewriting the complex number from rectangular to polar form. 

3. The image below shows polar form complex number in which will be rewritten in rectangular form. Each piece is transformed and then added up. Finally, they will transformed back to polar form and taking the square root. It is easier to to do the square root when in polar form and easier in rectangular form when adding.

Figure 4. The image below shows the order in which we evaluate the complex numbers. 

4. We are given a phasor and must find the sinusoidal representation of the phasors. In order to this, we need to convert the phasor in polar form and then multiply as seen below. Finally, we can convert it to a time domain.

Figure 5. Transforming a phasor into a time domain.

5. We are told to find a current trough a .1H inductor with a sinusoidal voltage. In order to achieve this, we must know the frequency domain for an inductor.  and transforming the equations into polar form. When in polar form, we can find the current by dividing the polar form complex number. It is also easier to have this in polar form since it is easier to divide. Finally, it is then transformed to a time domain.

Figure 6. Finding the current in the form a time domain given a sinusoidal voltage and inductance. 

Phasors: Passive RL Circuit Response Lab Procedures and Results:
1. The lab involves measuring the gain and phase response of an RL circuit comparing them our theoretical values. As seen in figure 8, we calculated our theoretical gain difference and phase difference. We calculated a phase difference of 45 degrees for corner frequency. The gain difference=.015. We also calculated the high frequency gain=.02117 and the low frequency gain=.002117. Most importantly, we calculated the cut of frequency = 47000 rad/sec. Though we must keep in mind that we need to divide by 2π. so the cut of frequency in hertz is, Cut of frequency = 7480 Hz. This value will be used for the lab when entering the sinusoidal voltage of 1 v with the frequencies we are told to use.

Figure 7. The actual circuit with equations of gain and phase difference

Figure 8.  Measurements for cut of frequency.

2. We measure the actual resistance of the resistor R =47.5Ω and the conductor is assumed to be L=.001H since there is not way of measuring it. The actual set can be seen in figure 9 as seen below.

Figure 9. Actual circuit set up 

3. We will now measure the input voltage, current, and voltage across the inductor. From the graph values collected we will calculate the Gain and phase at each frequency. the vales can be measure by looking at figure 10 and 11.

Figure 10. Measuring the gain.

Figure 11. Measuring the phase.

4. The results can be seen below with the help of figure 10 and 11:

GAIN = amplitude of I/amplitude of V, PHASE = ∆T/T * 360

Figure 12. LOW FREQUENCY

Figure 13. HIGH FREQUENCY

5. The results can be seen below for theoretical and experimental values

Figure 14. Experimental and theoretical values

Learning Outcome:

The objective of the lab is to compare our experimental and theoretical results for gain and phase change. Our theoretical results are: cut off frequency, ωc = 47000rad/sec, gain = .02117 at low frequency, gain = .002117 at high frequency, gain = .015 at corner, phase shift = 5.71° at low frequency, phase shift = 84.3° at high frequency, phase shift = 45° at corner frequency. It is important to note when entering the frequency for the input sinusoidal voltage of 1v, we converted the cut off frequency into herts by dividing the value by 2π. So, the frequency in hertz is ωc=7480Hz. The input voltage frequencies were ω=ωc/10=748Hz(LOW), ω=ωc*10=74.8kHz(HIGH), ω=ωc=7480Hz(CORNER). We now compared with our experimental measurements by following figure 10 and 11 where gain = amplitude of voltage out/ amplitude of voltage in. Phase is calculated by the change in period from voltage input and voltage output divided by the period of the voltage input times 360. We acquired a gain of .02157 and phase of 7.326 at low frequency. The percent error for gain was 1.89% and shift was 28.2%. This shows that there is a close comparison between theoretical and experimental with a small difference in error. However, the phase shift was not accurately calculated since it was eyeballed using the picture. Unfortunately, due to time, we were unable to measure the high and corner results.