Carefully Read Section 5

Notes:

• Start this project as early as possible! Don’t wait until a day before the deadline!
• Each student must turn in a separate report.
• The lab section can be done in groups of TWO or fewer! Only TWO people can use the same plots from the Scope or Spectrum Analyzer. You must specify the name of your Lab partner on your individual report.
• Please try to reduce the size of plots and use tables to avoid wasting papers. Double-sided reports will be ok to submit.
• Carefully read section 5.5.
• I have provided some references at the end of this document. Please consider reading about Spectrum Analyzers to learn about how they operate.

Fourier Transform:

Assume that we like to design an LRC circuit similar as shown below. Let L=0.1H, R=1 ohm, C=4mF

1-Using differential equations write the loop equation for this circuit.

2-Find the frequency response function for this system.

3-Express the Fourier transform in polar form.

4-Sketch the bode plots for both magnitude and phase angle of the system above.

5-What is the center frequency of this circuit?

6-What is the bandwidth of this circuit?

7-Explain what happens to your bode plot as you change the value of L, R, and C.

8-Which components are critical in setting the center frequency?

9-What type of system is this and what is it used for?

10-Let’s assume we want to use this circuit to set my radio to receive 90.1 FM station ( What should be the center frequency of my radio receiver?

11-Change the values of the circuit above such that the center frequency is set to KPFA radio.

12-Using the bode plot verify that your answer above is correct. Show the plot.

13-What is the bandwidth of your new receiver?

14-Assuming the received signal is mixed with a white noise uniformly distributed from 1 Hz to 10 MHz. Do you think in the presence of such white noise I can still clearly listen to my favorite programs on KPFA radio? Explain your answer.

15-Carefully read about Poles and Zeros. Review Reference 5 of this document.

16-For the LRC circuit above (L=0.1H, R=1 ohm, C=4mF), express the s-domaintransfer function, H(s), andfind all the poles and zeros.

17-Using the Example in Reference 5 of this document, explain how you can (manually) sketch the magnitude verses frequency.

18-Using Simulink, verify your sketch above. You must show your circuit and the frequency response.
Lab Experiment:

For this part you need to use the lab setup in Salazar 2003. Review the provided references below and learn about how to use Spectrum Analyzers. Connect the equipments as shown below:

1-Disconnect the HP 33360C. Using the BK-Precise generate a f=500 KHz, 2Vpp sine waveform. Connect the signal to the Spectrum Analyzer and Scope, as shown above. Set the following on the Spectrum Analyzer: Center Frequency=500 KHz, Span=100KHz, Amp=10dB. Observe the frequency spectral of the signal. Slowly change the frequency of the signal and describe what is happening. Include a picture of the signal as it appears on the Spectrum Analyzer.

2-Search and observe if there are any frequency components at 2f, 3f, 4f, and 5f. If so, record the amplitude in Volts. Describe if they have any significance.

3-Using the Scope, set the Math Function to FFT. Change Time Division size to 200 micro seconds and find all the main frequency peaks. Take a picture of your results.

4-Show the expression for Fourier Transform of the generated signal. Explain if your mathematical expression is consistent with your observations from the Spectrum Analyzer and Scope.

5-At this point change the sine waveform to square waveform (f=500 KHz, 2Vpp). Find the first five harmonics on the Spectrum Analyzer and clearly draw them. You must clearly show the frequency and amplitude of each harmonic.

6-Slowly change the frequency of the signal and describe what is happening.

7-Find the first five harmonics on the Scope. Take a picture of the plot as shown on the scope.

8-Using Matlab verify your observation on the Scope. You must include the Matlab expression and plot and clearly show how they are consistent with what you see in the lab.

9-What happens as you change the duty cycle of the square waveform?

10-Connect the HP 33360C. At this point, assume the original signal is being interfered with an unknown signal (actually, a 945 KHz sinusoidal signal with 1dBm amplitude from the HP33360C). Observe the resulting signal on the Scope (time domain). Explain what is happening. What happens as you change the amplitude of the interfering signal?

11-Using the Spectrum analyzer or Scope (FFT mode) can you identify the exact frequency of the interfering signal? Include a picture from the Spectrum Analyzer or the Scope.

12-Using the T-connector, are you multiplying or summing the interfering and the original signal? Explain your answer using Fourier Transfer expressions.

References

1-Waveforms and harmonics:

2-Read about Spectrum Analyzers

3-How to use Spectrum Analyzewrs:

4-Spectrum Analyzer Tutorial:

5-Read about Poles and Zeros:

6-Read about dBm:

Lab Setup