Lab report 7 PDF

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Lab 7 Electrical Resonance (R-L-C series circuit) Phys 208

Introduction: This lab was designed to show us the way a circuit operates in reaction to alternating current (AC). In other words, we were supposed to see the sinusoidal appearance of the voltage across certain designated points across the R-L-C circuit. That said, electrical resonance in the circuit was the ultimate goal that was meant for us to understand and its effects to see. In addition, some of the real word applications of these lab could be in electrical circuits, where it is used to tune to specific frequencies.

Objectives: As stated above, the main purpose of this lab was to understand electrical resonance in an RLC circuit. In addition, it intends to familiarize us with the concept of alternating current in a circuit and how that is different with a simple DC circuits.

Procedure: Part 1: setting up the circuit This part is solely dedicated to getting to set up the circuit. The circuit includes a resistor, inductor, and a capacitor. It is connected to the ground, on one side, and to a source of signal, on the other. Moreover, as instructed we had to make certain adjustments on the part of the oscilloscope, in order to be able to get a clear graph on the screen. Below I have recorded properties of the circuit which was used in this lab, as instructed:

R = 10 Ohms, RL = 94.8 Ohms, C = 0.022 MicroFarads, L = 0.032 H

Part 2: Again, at this part of the lab, as instructed, we changed the frequency to 6 kHz. Thereafter we had to measure the period of the sinusoidal function for voltage that was being shown on the oscilloscope. We measured T to be: T = 0.0001575 s Using the fact that frequency is the inverse of period, we found f to be: f = 6.3492 kHz At this point we must calculate the percentage error: Error = (6.3492 – 6)/ 6 * 100 = 5.820% As we can see the error in the measured frequency relative to the set frequency is 5.820%.

Part 3: At this stage, we adjusted the magnitude of the voltage to be 0.5 V by centering the sine function vertically in the oscilloscope. At this as instructed we obtained a sketch of the graph, the amplitude which we had, the peak to peak voltage, and the rms voltage, all which we have provided below.

Vm = 0.5 v, peak to peak voltage = 1 v, V rms = 0.3535 v

Part 4: A) At this point, we changed the settings of the oscilloscope so that the traces of both channels would be visible. Then we were told to increase the frequency of the signal generator from 2 kHz to 20 kHz and observe the changes in the phases between the two signals, the size of the resistor signal, and the slight variation in generator output on CH1. Below, we have provided a qualitative description of our observations: As we increased the frequency the graphs would go more out of phase. Meanwhile, the amplitude o the 2 nd signal remains the same. However, the generator output on channel 1 increases at first, and then decreases after 7 kHz. B) We, then, were told to adjust the frequency until the two signals were in phase. Doing so, we obtained the corresponding frequency to be 6 kHz. This frequency is also called the resonance frequency.

Part 5: In this part, we changed the display mode of the oscilloscope to XY, which meant that CH1 would be depicted on the horizontal axis and CH2 would be depicted on the vertical axis. We then, observed that the graph would portray an ellipse. This figure is called the Lissajous figure. We then scanned though the range of frequencies and noted down our observations as follows: The minor axis of the ellipse is increasing as we go to 6 kHz, while the major axis decreases. Thereafter, the major axis starts increasing, and the minor axis decreases, until the frequency gets to 20 and the ellipse is exactly the same size as it was on 2 kHz. Below we have made a rough sketch of the ellipse shown in the oscilloscope.

According to a property of a Lissajous figure, when the two signals are in phase, the pattern should become a straight line. Hence, we found this in-phase frequency to be 6.08 kHz.

Part 6: Impedance At this stage, we would leave the frequency at resonance and set V0 to 0.5 volts. Then we measure Vr on CH2. The corresponding value for Vr is 0.352 volts. Then, we use the data we have to calculate I0 and use that to calculate the impedance. Io = VR/ R -> .352/(94.8 + 10) = 0.003358, Z = V o / Io - > Z = 148.8981 Ohms

Part 7: Taking a resonance curve In this part we will measure pairs of values of V0 and Vr at various frequencies, so as to be able to plot a resonance curve. A resonance curve is a plot of I0/V0 vs. frequency.

f

Vo

Vr

Io

Io/Vo

2

0.484

0.092

8.518e-4

0.00176

3

0.48

0.148

0.00137

0.00285

4

0.472

0.236

0.00218 5

0.00462 9 0.00696

5

0.452

0.34

0.00314 8

5.2

0.44

0.348

0.00322

0.00732 3

5.4

0.44

0.356

0.00329 6

0.00749 1

5.6

0.436

0.36

0.00333

0.00764 5

5.8

0.432

0.356

0.00329 6

0.00763 0

6

0.432

0.352

0.00325 9

0.00754 4 0.00715 4

6.2

0.44

0.34

0.00314 8

6.4

0.44

0.332

0.00307

0.00698 6 0.00658 9

6.6

0.444

0.316

0.00292 5

6.8

0.452

0.304

0.00281 4

0.00622 7

7

0.456

0.292

0.00270 3

0.00592 9

8

0.46

0.232

0.00214 8

0.00466 9

9

0.472

0.188

0.00174 0

0.00368 8

Part 8: measuring phase shift Unfortunately we did not have time to complete this part.

Analysis: 1) Below, I have calculated the resonance frequency of the R-L-C series circuit, using the values on the bottom of the circuit board. w= 1/sqrt(LC) = 37.6889 rad/s  f resonance = w / 2*pi = 5.998 kHz comparing this measure frequency with those obtained in parts 4b and 5 which are, respectively, 6 kHz and 6.08 kHz, we observe that the error is very negligible and thus our calculations and measurements do indeed match. The theoretical value for impedance is 108 Ohms, whereas the measured value for impedance at resonance is 148 Ohms, which is actually a pretty large difference.

2) At frequency 4 kHz, the impedance is V0/I0 = 216 Ohms. The value of the impedance calculated from the circuit values is 210 Ohms, which was obtained using the formula Z = Sqrt(r^2 + (wL – 1/wC)^2) 3) At this point, we have plotted I0/V0 against frequency.

I0/V0 vs. frequency 0.01 0.01 0.01 0.01 0.01 0 0 0 0 0

1

2

3

4

5

6

7

8

9

f1 = 5, f2 = 5.2. Q = fo / (f2 – f1) -> 6 / (5.2-5) = 30. Theoretical = 0.01117. The values differ greatly.

Report Questions: Answer to question 1: At resonance the impedance, Z is at its minimum value, ( =R ). Therefore, the circuit current at this frequency will be at its maximum value of V/R.

Conclusion:

In this lab, we were able to analyze RLC circuits as Alternating current was applied to the circuit instead of DC. We used different methods to calculate or measure the frequency, especially at resonance. That sad, we also got to find the resonance frequency with the use of Lissahous figure and also with the use of pre-known formulas. Most importantly though, we got a parabolic curve for the graph. The resonance was near what we calculated in regards to the theoretical value. A source of error is not accurately recording that voltages as frequency is changed. Another source of error is the not counting the divisions on the oscilloscope closely enough....