Frequency Response and Filters Objectives: This experiment provides practical experiences with frequency responses of analog filters. Filters will be constructed and graphs of gain magnitude and phase versus frequency will be created based on data collected in the lab. Pre Lab: Read and understand Hambley Chapter 6 Sections 6.1, 6. and 6.5 covering low pass filters and high pass filters. The gain (also called the transfer function) of a filter is the ratio of the phasor output voltage to the phasor input voltage. Using phasor analysis, the complex gain of a first-order RC low pass filter as a function of radian frequency is given by the equations below. From the Gain equation, the Gain magnitude can be determined as shown below in the Gain equation and the phase can be determined as shown below in the Phase equation. V Gain = V 1 1/jωC = = R + 1/jωC 1 + 1 jωrc Gain = 1 1 + ( ωrc ) phase = arctan( ωrc) Use nominal values of your resistor and capacitor components shown in Figure 1 to determine theoretical gain magnitude and phase at frequencies of 50 Hz, 00 Hz, 1kHz, and 5 khz. These values will be used to determine if your circuit is correct in the lab. Show all calculations in your lab report. Perform phasor analysis to determine the transfer function for the high pass filter circuit shown in Figure. Use the measured values of your resistor and capacitor components to determine theoretical gain and phase at frequencies of 500 Hz, 1kHz, 5kHz, and 50 khz Compare these values with what you recorded in Part B Table in lab during the first week of lab. Show all calculations in your lab report. Do the experimental values appear to be correct? Mark the four points calculated in the Prelabs (for the low pass and high pass filters) on the graphs for Parts A and B. Does your experimental values correlate with what is expected? Part A: Low Pass Filter Frequency Response Procedure: 1. Measure the 1 kω resistor using the Fluke 45 as an Ohmmeter. Measure the 0.8 µf capacitor using the universal bridge on the instructor s table. Enter these values on the data sheet.. Construct the low pass filter circuit illustrated in Figure 1. It is convenient to use banana leads for the component connections, but we suggest using probes for the scope measurements (remember to check that the amplification factor for CH1 and CH are 10x). When using the Page 1 of 7
probes, use common sense precautions to avoid damaging the sensitive contacts and clips. 3. Using any method, set the waveform generator output to a sinusoid with an amplitude of 1 V peak-to-peak and an initial frequency of 50 Hz. 4. The oscilloscope display may have a natural coarseness, which can be reduced by selecting the Average function. Try to use as few samples as possible to avoid long delays while moving from one display to another. Set both channels to the same Volts/Division setting. 5. Figure 1: The Low Pass Filter Circuit 1µF NOTE: If the displayed wave is extremely jumpy, the problem is generally faulty banana leads or an ungrounded probe. If you cannot remedy the problem, ask for assistance. Push the AUTOSET button to display both channels. Reposition so that the 0V is at the midline on the both channels and the waveforms appear to overlap. Adjust the Volts/division to 00mV/division for both channels and the sec/division to display more then one complete period. 6. Press the MEASURE button to determine the Voltage pk-pk amplitudes for both channels, and record in the data sheet for the frequency of 50 Hz. At least one full period of the waveform must be in view to for the oscilloscope measure functions to be accurate. 7. Press the CURSOR button and adjust the cursors to measure the change in time between peaks. Record the t in the data sheet for the frequency of 50 Hz. 8. Calculate the Gain magnitude and Phase using the following formulas and enter in Table 1 of the data sheet. VOut Gain = Phase = Frequency x t x 360 VIn 9. Compare with what you would expect for this circuit based on theoretical calculations from the prelab. 10. Repeat steps 6 through 8 to make measurements for all frequencies shown in Table 1 and record these values in Table 1. 11. Graph the Gain and Phase on the semi log paper provided. Do your results agree with what you would expect for a low pass filter? Explain what is meant by a low pass filter. Part B: High Pass Filter Frequency Response Page of 7
Procedure: 1. Measure the 1-kΩ resistor using the Fluke 45 as an Ohmmeter. Measure the 0.1-µF capacitor using the universal bridge on the instructor s table. Enter these values on the data sheet.. Construct the high pass filter circuit illustrated in Figure. It is convenient to use banana leads for the component connections, but we suggest using probes for the scope measurements (remember to check that the amplification factor for CH1 and CH are 10x). When using the probes, use common sense precautions to avoid damaging the sensitive contacts and clips. 3. Using any method, set the waveform generator output to a sinusoid with an amplitude of 1 V peak-to-peak and an initial frequency of 50 Hz. 4. The oscilloscope display may have a natural coarseness, which can be reduced by selecting the Average function. Try to use as few samples as possible to avoid long delays while moving from one display to another. Set both channels to the same Volts/Division setting. Figure : The High Pass Filter Circuit NOTE: If the displayed wave is extremely jumpy, the problem is generally faulty banana leads or an ungrounded probe. If you cannot remedy the problem, ask for assistance. 5. Push the AUTOSET button to display both channels. Reposition so that the 0V is at the midline on the both channels and the waveforms appear to overlap. Adjust the Volts/division to 00mV/division for both channels and the sec/division to display more than one complete period. 6. Press the MEASURE button to determine the Voltage pk-pk amplitudes for both channels, and record in the data sheet for the frequency of 50 Hz. At least one full period of the waveform must be in view to for the oscilloscope measure functions to be accurate. 7. Press the CURSOR button and adjust the cursors to measure the change in time between peaks. Record the t in the data sheet for the frequency of 50 Hz. 8. Calculate the Gain magnitude and Phase using the following formulas and enter in Table of the data sheet. VOut Gain = Phase = Frequency x t x 360 VIn 9. Repeat steps 6 through 8 to make measurements for all frequencies shown in Table and record these values in Table. Page 3 of 7
10. Graph the Gain and Phase on the semi log paper provided. Do your results agree with what you would expect for a high pass filter? Explain what is meant by a high pass filter. Part C: Using Filters To Isolate Frequencies In An AM Signal Procedure: 1. For this part, you will use the function generator to create an amplitude modulated (AM) signal to simulate the sound of an engine running at 6000 RPM with an associated high-frequency vibration. The AM signal is given by v ( t) = A[1 + µ cos( ω t)]cos( ω t) AM m c Expanding the equation and using the trigonometric identity for the sum of cosines, we have: Aµ Aµ vam ( t) = Acos( ωct) + cos[( ω m + ωc ) t] + cos[( ω m ωc ) t] in which A is the carrier amplitude which we will set at 1 V, µ is the modulation index which we choose as 0.3, cos(ω m t)is the modulating waveform which represents a high-frequency (5 khz) vibration, and cos(ω c t) is the carrier waveform that results from engine rotation. For an engine speed of 6000 rpm, the corresponding carrier frequency is 100 Hz. Notice that the AM signal has three components with frequencies of 100 Hz, 4900 Hz and 5100 Hz. The 100 Hz component results from engine rotation while the 4900 and 5100 Hz components result from vibration. We will use a low pass filter to pass the 100 Hz component and reject the vibration components. Then we will use a high pass filter to pass the vibration components and reject the 100 Hz component.. Set the waveform generator to sinusoidal mode. Switch to the AM mode by pressing the SHIFT then AM buttons. Press the FREQ button and enter a frequency of 100 Hz for the carrier waveform. Press the AMPL button and enter 1 V pk-pk for the amplitude of the carrier waveform. Press the SHIFT then AMPL buttons to set the DEPTH to 30%. This corresponds to µ = 0.3. Press the SHIFT then FREQ buttons to set the frequency of the modulating waveform and enter 5 khz. 3. Connect the low pass filter and oscilloscope as illustrated in Figure 1. Turn off Channel of the oscilloscope and examine the AM signal as the input to the filter. Adjust the volts/division and seconds/division so that one period fills the screen. 4. Turn on Channel so that one complete waveform is viewed on the oscilloscope. At least one full period of the waveform must be in view to for the oscilloscope measure functions to be accurate. Using the Wavestar PC software print the resulting display. Examine the phase, amplitude, and appearance of the signal that is the output of the low pass filter. Explain in your lab report the effect that the low pass filter has on the AM signal. Could this filtered signal be used to determine the engine RPM? 5. Disconnect the low pass filter and then connect the high pass filter and oscilloscope as illustrated in Figure. Examine the resulting waveforms for the filter input and output. Using Wavestar PC software print the resulting display. Examine the appearance of the signal that is the output of the high pass filter. Explain in your lab report the effect that the high pass filter has on the AM signal. Page 4 of 7
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Part A: Low Pass Filter Measurements Nominal 1kΩ resistor value Nominal 0.8µF capacitor value Table 1 f (Hz) CH-1 pk-pk CH- pk-pk t gain phase "+" if CH- leads 50 100 00 500 800 1k k 5k 10k Part B: High Pass Filter Measurements Nominal 1kΩ resistor value Nominal 0.1 µf capacitor value Table f (Hz) CH-1 pk-pk CH- pk-pk t gain phase "+" if CH- leads 50 100 00 500 800 1k k 5k 10k Your Name Lab Instructor Page 7 of 7