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    • NATIONAL COLLEGE OF SCIENCE AND TECHNOLOGY Amafel Bldg. Aguinaldo Highway Dasmariñas City, Cavite EXPERIMENT 5 “Fourier Theory – Frequency Domain and Time Domain”Pula, Rolando A. September 06, 2011Signal Spectra and Signal Processing/BSECE 41A1 Score: Engr. Grace Ramones Instructor
    • OBJECTIVES: 1. Learn how a square wave can be produced from a series of sine waves at different frequencies and amplitudes. 2. Learn how a triangular can be produced from a series of cosine waves at different frequencies and amplitudes. 3. Learn about the difference between curve plots in the time domain and the frequency domain. 4. Examine periodic pulses with different duty cycles in the time domain and in the frequency domain. 5. Examine what happens to periodic pulses with different duty cycles when passed through low-pass filter when the filter cutoff frequency is varied.
    • SAMPLE COMPUTATIONDUTY CYCLEFREQUENCYFIRST ZERO CROSSING POINTBANDWIDTHBW =
    • DATA SHEET:Materials:One function generatorOne oscilloscopeOne spectrum analyzerOne LM 741 op-ampTwo 5 nF variable capacitorsResistors: 5.86 kΩ, 10 kΩ, and 30 kΩTheory:Communications systems are normally studies using sinusoidal voltage waveforms to simplify the analysis.In the real world, electrical information signal are normally nonsinusoidal voltage waveforms, such as audiosignals, video signals, or computer data. Fourier theory provides a powerful means of analyzingcommunications systems by representing a nonsinusoidal signal as series of sinusoidal voltages addedtogether. Fourier theory states that a complex voltage waveform is essentially a composite of harmonicallyrelated sine or cosine waves at different frequencies and amplitudes determined by the particular signalwaveshape. Any, nonsinusoidal periodic waveform can be broken down into sine or cosine wave equal to thefrequency of the periodic waveform, called the fundamental frequency, and a series of sine or cosine wavesthat are integer multiples of the fundamental frequency, called the harmonics. This series of sine or cosinewave is called a Fourier series.Most of the signals analyzed in a communications system are expressed in the time domain, meaning thatthe voltage, current, or power is plotted as a function of time. The voltage, current, or power is representedon the vertical axis and time is represented on the horizontal axis. Fourier theory provides a new way ofexpressing signals in the frequency domain, meaning that the voltage, current, or power is plotted as afunction of frequency. Complex signals containing many sine or cosine wave components are expressed assine or cosine wave amplitudes at different frequencies, with amplitude represented on the vertical axis andfrequency represented on the horizontal axis. The length of each of a series of vertical straight linesrepresents the sine or cosine wave amplitudes, and the location of each line along the horizontal axisrepresents the sine or cosine wave frequencies. This is called a frequency spectrum. In many cases thefrequency domain is more useful than the time domain because it reveals the bandwidth requirements of thecommunications system in order to pass the signal with minimal distortion. Test instruments displayingsignals in both the time domain and the frequency domain are available. The oscilloscope is used to displaysignals in the time domain and the spectrum analyzer is used to display the frequency spectrum of signals inthe frequency domain.In the frequency domain, normally the harmonics decrease in amplitude as their frequency gets higher untilthe amplitude becomes negligible. The more harmonics added to make up the composite waveshape, themore the composite waveshape will look like the original waveshape. Because it is impossible to design acommunications system that will pass an infinite number of frequencies (infinite bandwidth), a perfectreproduction of an original signal is impossible. In most cases, eliminate of the harmonics does notsignificantly alter the original waveform. The more information contained in a signal voltage waveform (afterchanging voltages), the larger the number of high-frequency harmonics required to reproduce the originalwaveform. Therefore, the more complex the signal waveform (the faster the voltage changes), the wider thebandwidth required to pass it with minimal distortion. A formal relationship between bandwidth and theamount of information communicated is called Hartley’s law, which states that the amount of informationcommunicated is proportional to the bandwidth of the communications system and the transmission time.Because much of the information communicated today is digital, the accurate transmission of binary pulsesthrough a communications system is important. Fourier analysis of binary pulses is especially useful incommunications because it provides a way to determine the bandwidth required for the accurate
    • transmission of digital data. Although theoretically, the communications system must pass all the harmonicsof a pulse waveshape, in reality, relatively few of the harmonics are need to preserve the waveshape.The duty cycle of a series of periodic pulses is equal to the ratio of the pulse up time (t O) to the time period ofone cycle (T) expressed as a percentage. Therefore,In the special case where a series of periodic pulses has a 50% duty cycle, called a square wave, the plot inthe frequency domain will consist of a fundamental and all odd harmonics, with the even harmonics missing.The fundamental frequency will be equal to the frequency of the square wave. The amplitude of each oddharmonic will decrease in direct proportion to the odd harmonic frequency. Therefore,The circuit in Figure 5–1 will generate a square wave voltage by adding a series of sine wave voltages asspecified above. As the number of harmonics is decreased, the square wave that is produced will have moreripples. An infinite number of harmonics would be required to produce a perfectly flat square wave.Figure 5 – 1 Square Wave Fourier Series XSC1 Ext T rig V6 + R1 J1 _ A B 10.0kΩ + _ + _ 10 V Key = A V1 R2 J2 10 Vpk 10.0kΩ 1kHz Key = B 0° V2 R3 J3 4 155 0 8 10 14 13 12 6 R7 102 09 3 100Ω 3.33 Vpk 10.0kΩ 3kHz Key = C 0° V3 R4 J4 2 Vpk 10.0kΩ 5kHz Key = D 0° V4 R5 J5 1.43 Vpk 10.0kΩ 7kHz 0° Key = E V5 J6 R6 1.11 Vpk 10.0kΩ 9kHz Key = F 0° .The circuit in Figure 5-2 will generate a triangular voltage by adding a series of cosine wave voltages. Inorder to generate a triangular wave, each harmonic frequency must be an odd multiple of the fundamentalwith no even harmonics. The fundamental frequency will be equal to the frequency of the triangular wave,the amplitude of each harmonic will decrease in direct proportion to the square of the odd harmonicfrequency. Therefore,Whenever a dc voltage is added to a periodic time varying voltage, the waveshape will be shifted up by theamount of the dc voltage.
    • Figure 5 – 2 Triangular Wave Fourier Series XSC1 Ext T rig V6 + R1 J1 _ A B 10.0kΩ + _ + _ 15 V Key = A V1 R2 J2 10 Vpk 10.0kΩ 1kHz 90° V2 Key = B R3 J3 13 12 1 2 3 4 5 8 9 11 0 R7 6 0 1.11 Vpk 100Ω 10.0kΩ 3kHz 90° V3 Key = C R4 J4 0.4 Vpk 10.0kΩ 5kHz 90° V4 Key = D R5 J5 0.2 Vpk 10.0kΩ 7kHz 90° Key = EFor a series of periodic pulses with other than a 50% duty cycle, the plot in the frequency domain will consistof a fundamental and even and odd harmonics. The fundamental frequency will be equal to the frequency ofthe periodic pulse train. The amplitude (A) of each harmonic will depend on the value of the duty cycle. Ageneral frequency domain plot of a periodic pulse train with a duty cycle other than 50% is shown in thefigure on page 57. The outline of peaks if the individual frequency components is called envelope of thefrequency spectrum. The first zero-amplitude frequency crossing point is labelled f o = 1/to, there to is the uptime of the pulse train. The first zero-amplitude frequency crossing point f o) determines the minimumbandwidth (BW0 required for passing the pulse train with minimal distortion.Therefore, A f=1/to 2/to f Frequency Spectrum of a Pulse Train
    • Notice than the lower the value of to the wider the bandwidth required to pass the pulse train with minimaldistortion. Also note that the separation of the lines in the frequency spectrum is equal to the inverse of thetime period (1/T) of the pulse train. Therefore a higher frequency pulse train requires a wider bandwidth (BW)because f = 1/TThe circuit in Figure 5-3 will demonstrate the difference between the time domain and the frequency domain.It will also determine how filtering out some of the harmonics effects the output waveshape compared to theoriginal3 input waveshape. The frequency generator (XFG1) will generate a periodic pulse waveform appliedto the input of the filter (5). At the output of the filter (70, the oscilloscope will display the periodic pulsewaveform in the time domain, and the spectrum analyzer will display the frequency spectrum of the periodicpulse waveform in the frequency domain. The Bode plotter will display the Bode plot of the filter so that thefilter bandwidth can be measured. The filter is a 2-pole low-pass Butterworth active filter using a 741 op-amp.Figure 5-3 Time Domain and Frequency Domain XFG1 XSC1 C1 XSA1 Ext T rig + 2.5nF 50% _ Key=A A _ B _ IN T + + R1 R2 741 30kΩ 30kΩ 42 OPAMP_3T_VIRTUAL 0 6 0 31 R3 C2 R4 5.56kΩ 10kΩ XBP1 2.5nF 50% Key=A R5 IN OUT 10kΩProcedure:Step 1 Open circuit file FIG 5-1. Make sure that the following oscilloscope settings are selected: Time base (Scale = 200 µs/Div, Xpos = 0, Y/t), Ch A (Scale = 5V/Div, Ypos = 0, DC), Ch B (Scale = 50 mV/Div, Ypos = 0, DC), Trigger (Pos edge, Level = 0, Auto). You will generate a square wave curve plot on the oscilloscope screen from a series of sine waves called a Fourier series.Step 2 Run the simulation. Notice that you have generated a square wave curve plot on the oscilloscope screen (blue curve) from a series of sine waves. Notice that you have also plotted the fundamental sine wave (red). Draw the square wave (blue) curve on the plot and the fundamental sine wave (red) curve plot in the space provided.
    • Step 3 Use the cursors to measure the time periods for one cycle (T) of the square wave (blue) and the fundamental sine wave (red) and show the value of T on the curve plot. T1 = 1.00 MS T2 = 1.00 MSStep 4 Calculate the frequency (f) of the square wave and the fundamental sine wave from the time period. F = 1 KHZQuestions: What is the relationship between the fundamental sine wave and the square wave frequency (f)? THEY ARE THE EQUAL.What is the relationship between the sine wave harmonic frequencies (frequencies of sine wave generatorsf3, f5, f7, and f9 in figure 5-1) and the sine wave fundamental frequency (f 1)? THE FREQUENCIES ARE ALL ODD MULTIPLES.What is the relationship between the amplitude of the harmonic sine wave generators and the amplitude ofthe fundamental sine wave generator? THE AMPLITUDE OF THE ODD HARMONICS DECREASE IN DIRECT PROPORTION TO ODD HARMONIC FREQUENCY.Step 5 Press the A key to close switch A to add a dc voltage level to the square wave curve plot. (If the switch does not close, click the mouse arrow in the circuit window before pressing the A key). Run the simulation again. Change the oscilloscope settings as needed. Draw the new square wave (blue) curve plot on the space provided.Question: What happened to the square wave curve plot? Explain why. IT MOVED UPWARD. IT IS BECAUSE DC VOLTAGE ADDED.Step 6 Press the F and E keys to open the switches F and E to eliminate the ninth and seventh harmonic sine waves. Run the simulation again. Draw the new curve plot (blue) in the space provided. Note any change on the graph.Step 7 Press the D key to open the switch D to eliminate the fifth harmonics sine wave. Run the simulation again. Draw the new curve plot (blue) in the space provided. Note any change on the graph.
    • Step 8 Press the C key to open switch C and eliminate the third harmonic sine wave. Run the simulation again.Question: What happened to the square wave curve plot? Explain. IT BECAME SINUSOIDAL WAVE. ALL THE HARMONICS ARE ALL GONE SO IT APPEARS LIKE THE FUNDAMENTAL SINE WAVE.Step 9 Open circuit file FIG 5-2. Make sure that the following oscilloscope settings are selected: Time base (Scale = 200 µs/Div, Xpos = 0, Y/t), Ch A (Scale = 5V/Div, Ypos = 0, DC), Ch B (Scale = 100 mV/Div, Ypos = 0, DC), Trigger (Pos edge, Level = 0, Auto). You will generate a triangular wave curve plot on the oscilloscope screen from a series of sine waves called a Fourier series.Step 10 Run the simulation. Notice that you have generated a triangular wave curve plot on the oscilloscope screen (blue curve) from the series of cosine waves. Notice that you have also plotted the fundamental cosine wave (red). Draw the triangular wave (blue) curve plot and the fundamental cosine wave (red) curve plot in the space provided.Step 11 Use the cursors to measure the time period for one cycle (T) of the triangular wave (blue) and thefundamental (red), and show the value of T on the curve plot. T1 = 1.00 MS T2 = 1.00 MSStep 12 Calculate the frequency (f) of the triangular wave from the time period (T). F = 1 KHZQuestions: What is the relationship between the fundamental frequency and the triangular wave frequency? THEY ARE THE EQUAL.What is the relationship between the harmonic frequencies (frequencies of generators f 3, f5, and f7 in figure 5-2) and the fundamental frequency (f1)? THEY ARE ALL ODD FUNCTIONS.What is the relationship between the amplitude of the harmonic generators and the amplitude of thefundamental generator? THE AMPLITUDE OF THE HARMONIC GENERATORS DECREASES IN DIRECT PROPORTION TO THE SQUARE OF THE ODD HARMONIC FREQUENCY
    • Step 13 Press the A key to close switch A to add a dc voltage level to the triangular wave curve plot. Run the simulation again. Draw the new triangular wave (blue) curve plot on the space provided.Question: What happened to the triangular wave curve plot? Explain. IT MOVED UPWARD. IT IS BECAUSE DC VOLTAGE ADDED.Step 14 Press the E and D keys to open switches E and D to eliminate the seventh and fifth harmonic sine waves. Run the simulation again. Draw the new curve plot (blue) in the space provided. Note any change on the graph.Step 15 Press the C key to open the switch C to eliminate the third harmonics sine wave. Run the simulation again.Question: What happened to the triangular wave curve plot? Explain. IT BECAME SINE WAVE. BECAUSE THE HARMONIC SINE WAVES ARE GONE, IT APPEARS AS A FUNDAMENTAL SINE WAVE.Step 16 Open circuit FIG 5-3. Make sure that following function generator settings are selected: Square wave, Freq = 1 kHz, Duty cycle = 50%, Ampl – 2.5 V, Offset = 2.5 V. Make sure that the following oscilloscope settings are selected: Time base (Scale = 500 µs/Div, Xpos = 0, Y/T), Ch A (Scale = 5 V/Div, Ypos = 0, DC), Ch B (Scale = 5 V/Div, Ypos = 0, DC), Trigger (pos edge, Level = 0, Auto). You will plot a square wave in the time domain at the input and output of a two- pole low-pass Butterworth filter.Step 17 Bring down the oscilloscope enlargement and run the simulation to one full screen display, then pause the simulation. Notice that you are displaying square wave curve plot in the time domain (voltage as a function of time). The red curve plot is the filter input (5) and the blue curve plot is the filter output (7)Question: Are the filter input (red) and the output (blue) plots the same shape disregarding any amplitudedifferences? YES.Step 18 Use the cursor to measure the time period (T) and the time (f o) of the input curve plot (red) and record the values. T= 1 MS TO = 500.477µS
    • Step 19 Calculate the pulse duty cycle (D) from the to and T D = 50.07%.Question: How did your calculated duty cycle compare with the duty cycle setting on the function generator? THE DIFFERENCE IS 0.07%.Step 20 Bring down the Bode plotter enlargement to display the Bode plot of the filter. Make sure that the following Bode plotter settings are selected; Magnitude, Vertical (Log, F = 10 dB, I = -40 dB), Horizontal (Log, F = 200 kHz, I = 100 Hz). Run the simulation to completion. Use the cursor to measure the cutoff frequency (fC) of the low-pass filter and record the value. FC = 21.197Step 21 Bring down the analyzer enlargement. Make sure that the following spectrum analyzer settings are selected: Freq (Start = 0 kHz, Center = 5 kHz, End = 10 kHz), Ampl (Lin, Range = 1 V/Div), Res = 50 Hz. Run the simulation until the Resolution frequencies match, then pause the simulation. Notice that you have displayed the filter output square wave frequency spectrum in the frequency domain, use the cursor to measure the amplitude of the fundamental and each harmonic to the ninth and record your answers in table 5-1. Table 5-1 Frequency (kHz) Amplitude f1 1 5.048 V f2 2 11.717 µV f3 3 1.683 V f4 4 15.533 µV f5 5 1.008 V f6 6 20.326 µV f7 7 713.390 mV f8 8 25.452 µV f9 9 552.582 mVQuestions: What conclusion can you draw about the difference between the even and odd harmonics for asquare wave with the duty cycle (D) calculated in Step 19? THERE IS ONLY ODD HARMONICS. THE EVEN HARMONICS IS ALMOST ZERO.What conclusions can you draw about the amplitude of each odd harmonic compared to the fundamental fora square wave with the duty cycle (D) calculated in Step 19? THE AMPLITUDE OF ODD HARMONICS DECREASES IN DIRECT PROPORTION WITH THE ODD HARMONIC FREQUENCY.Was this frequency spectrum what you expected for a square wave with the duty cycle (D) calculated in Step19? YES.Based on the filter cutoff frequency (f C) measured in Step 20, how many of the square wave harmonicswould you expect to be passed by this filter? Based on this answer, would you expect much distortion of theinput square wave at the filter? Did your answer in Step 17 verify this conclusion? THERE ARE SQUARE WAVES. YES, IT IS EXPECTED.Step 22 Adjust both filter capacitors (C) to 50% (2.5 nF) each. (If the capacitors won’t change, click the mouse arrow in the circuit window). Bring down the oscilloscope enlargement and run the simulation to one full screen display, then pause the simulation. The red curve plot is the filter input and the blue curve plot is the filter output.Question: Are the filter input (red) and output (blue) curve plots the same shape, disregarding any amplitudedifferences? NO.Step 23 Bring down the Bode plotter enlargement to display the Bode plot of the filter. Use the cursor to measure the cutoff frequency (Fc of the low-pass filter and record the value. FC = 2.12 KHZ
    • Step 24 Bring down the spectrum analyzer enlargement to display the filter output frequency spectrum in the frequency domain, Run the simulation until the Resolution Frequencies match, then pause the simulation. Use cursor to measure the amplitude of the fundamental and each harmonic to the ninth and record your answers in Table 5-2. Table 5-2 Frequency (kHz) Amplitude f1 1 4.4928 V f2 2 4.44397µV f3 3 792.585 mV f4 4 323.075 µV f5 5 178.663mV f6 6 224.681 µV f7 7 65.766 mV f8 8 172.430 µV f9 9 30.959 mVQuestions: How did the amplitude of each harmonic in Table 5-2 compare with the values in Table 5-1? THE AMPLITUDE IS LOWER COMPARE WITH THE THE VALUES IN TABLE 5-1.Based on the filter cutoff frequency (f c), how many of the square wave harmonics should be passed by thisfilter? Based on this answer, would you expect much distortion of the input square wave at the filter output?Did your answer in Step 22 verify this conclusion? THERE ARE LESS THAN 5 SQUARE WAVE HARMONICS. YES, IT IS EXPECTED TO HAVE MUCH DISTORTION OF THE INPUT SQUARE WAVE AT OUTPUT.Step 25 Change the both capacitor (C) back to 5% (0.25 nF). Change the duty cycle to 20% on the function generator. Bring down the oscilloscope enlargement and run the simulation to one full screen display, then pause the simulation. Notice that you have displayed a pulse curve plot on the oscilloscope in the time domain (voltage as a function of time). The red curve plot is the filter input and the blue curve plot is the filter output.Question: Are the filter input (red) and the output (blue) curve plots the same shape, disregarding anyamplitude differences? YES.Step 26 Use the cursors to measure the time period (T) and the up time (t o) of the input curve plot (red) and record the values. T= 1 MS TO =Step 27 Calculate the pulse duty cycle (D) from the to and T. D = 19.82%Question: How did your calculated duty cycle compare with the duty cycle setting on the function generator? THEIR DIFFERENCE IS 0.18%Step 28 Bring down the Bode plotter enlargement to display the Bode plot of the filter. Use the cursor to measure the cutoff frequency (fC) of the low-pass filter and record the value. FC = 21.197 KHZStep 29 Bring down the spectrum analyzer enlargement to display the filter output frequency spectrum in the frequency domain. Run the simulation until the Resolution Frequencies match, then pause the simulation. Draw the frequency plot in the space provided. Also draw the envelope of the frequency spectrum.
    • 5.041 kHzQuestion: Is this the frequency spectrum you expected for a square wave with duty cycle less than 50%? YES.Step 30 Use the cursor to measure the frequency of the first zero crossing point (f o) of the spectrum envelope and record your answer on the graph. FO = 5.041 KHZStep 31 Based on the value of the to measured in Step 26, calculate the expected first zero crossing point (fo) of the spectrum envelope. FO = 5.045 KHZQuestion: How did your calculated value of f o compare the measured value on the curve plot? THE DIFFERENCE IS 0.004 HZStep 32 Based on the value of fo, calculate the minimum bandwidth (BW) required for the filter to pass the input pulse waveshape with minimal distortion. BW = 5.045 kHzQuestion: Based on this answer and the cutoff frequency (f c) of the low-pass filter measure in Step 28, wouldyou expect much distortion of the input square wave at the filter output? Did your answer in Step 25 verifythis conclusion? NO, BASED ON THIS ANSWER AND THE CUTOFF FREQUENCY (FC) OF THE LOW- PASS FILTER MEASURE IN STEP 28, THERE IS LESS DISTORTION. YES, IT VERIFIED. THE HIGHER THE BANDWIDTH, THE LESSER THE DISTORTION.Step 33 Adjust the filter capacitors (C) to 50% (2.5 nF) each. Bring down the oscilloscope enlargement and run the simulation to one full screen display, then pause the simulation. The red curve plot is the filter input and the blue curve plot is the filter output.Question: Are the filter input (red) and the output (blue) curve plots the same shape, disregarding anyamplitude differences? NO.Step 34 Bring down the Bode plotter enlargement to display the Bode plot of the filter. Use the cursor tomeasure the cutoff frequency (fc) of the low-pass filter and record the value. FC = 4.239 KHZQuestions: Was the cutoff frequency (fc) less than or greater than the minimum bandwidth (BW) required topass the input waveshape with minimal distortion as determined in Step 32? THE FC IS GREATER THAN THE BW TO PASS THE INPUT WAVESHAPE WITH MINIMUM DISTORTIONBased on this answer, would you expect much distortion of the input pulse waveshape at the filter output?Did your answer in Step 33 verify this conclusion? NO, THERE WILL HAVE MUCH DISTORTION IN THE INPUT WAVESHAPE AT THE OUTPUT IF THE BANDWIDTH IS DEDUCTED.
    • Step 35 Bring down the spectrum analyzer enlargement to display the filter output frequency spectrum in the frequency domain. Run the simulation until the Resolution Frequencies match, then pause the simulation.Question: What is the difference between this frequency plot and the frequency plot in Step 29? THE DIFFERENCE IS THAT THE AMPLITUDE IS LOWER COMPARED WITH THE FREQUENCY PLOT IN STEP 29?
    • CONCLUSION Therefore, I can say that any nonsinusoidal waves can be a series of cosine or sine waves.It is consist of harmonics and the fundamental sine wave. A square wave is a series of odd sinefunctions while a triangular is a series of odd cosine wave. When a dc source is added, the curvewill move upward. When a duty cycle is 50%, there is only odd function, while duty cycle of lessthan 50% will have even and odd harmonics. The bandwidth is inversely proportional to the up timeof the pulse train and to the distortion of the input. The wider the bandwidth, the lesser thedistortion will be. The higher the increment of the capacitance the more the shape of the input andthe output will be different.