National college of science and technology
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The document describes an experiment to analyze active low-pass and high-pass filters. Students measured the gain-frequency response and cutoff frequencies of second-order Butterworth filters using op-amps. For the low-pass filter, the measured cutoff frequency was 5.321 kHz which closely matched the calculated value of 5.305 kHz. For the high-pass filter, the measured cutoff frequency of 5.156 kHz also closely matched the calculated value. When component values were changed significantly, the measured cutoff frequencies differed from calculations, likely due to op-amp bandwidth limitations. The results demonstrated second-order filters should roll off at -40 dB per decade as expected.
National college of science and technology
- 1. NATIONAL COLLEGE OF SCIENCE AND TECHNOLOGY Amafel Building, Aguinaldo Highway Dasmariñas City, Cavite Experiment No. 3 “ACTIVE LOW-PASS and ACTIVE HIGH-PASS FILTERS” Balane, Maycen M. July 14, 2011 Signal Spectra and Signal Processing/BSECE 41A1 Score: Engr. Grace Ramones Instructor
- 2. OBJECTIVES: 1. Plot the gain-frequency response and determine the cutoff frequency of a second-order (two-pole) low-pass active filter. 2. Plot the gain-frequency response and determine the cutoff frequency of a second-order (two-pole) high-pass active filter. 3. Determine the roll-off in dB per decade for a second-order (two-pole) filter. 4. Plot the phase-frequency response of a second-order (two-pole) filter.
- 3. SAMPLE COMPUTATIONS: Computed Av based on measured AdB AdB = 20 log A 4.006 = 20 log A Expected Av AdB = 20 log A 3.776 = 20 log A
- 4. DATA SHEET: THEORY In electronic communications systems, it is often necessary to separate a specific range of frequencies from the total frequency spectrum. This is normally accomplished with filters. A filter is a circuit that passes a specific range of frequencies while rejecting other frequencies. Active filters use active devices such as op-amps combined with passive elements. Active filters have several advantages over passive filters. The passive elements provide frequency selectivity and the active devices provide voltage gain, high input impedance, and low output impedance. The voltage gain reduces attenuation of the signal by the filter, the high input prevents excessive loading of the source, and the low output impedance prevents the filter from being affected by the load. Active filters are also easy to adjust over a wide frequency range without altering the desired response. The weakness of active filters is the upper-frequency limit due to the limited open-loop bandwidth (funity) of op-amps. The filter cutoff frequency cannot exceed the unity-gain frequency (funity) of the op-amp. Ideally, a high-pass filter should pass all frequencies above the cutoff frequency (fc). Because op-amps have a limited open-loop bandwidth (unity-gain frequency, funity), high-pass active filters have an upper-frequency limit on the high-pass response, making it appear as a band-pass filter with a very wide bandwidth. Therefore, active filters must be used in applications where the unity-gain frequency (funity) of the op-amp is high enough so that it does not fall within the frequency range of the application. For this reason, active filters are mostly used in low-frequency applications. The most common way to describe the frequency response characteristics of a filter is to plot the filter voltage gain (Vo/Vin) in dB as a function of frequency (f). The frequency at which the output power gain drops to 50% of the maximum value is called the cutoff frequency (fc). When the output power gain drops to 50%, the voltage gain drops 3 dB (0.707 of the maximum value). When the filter dB voltage gain is plotted as a function of frequency using straight lines to approximate the actual frequency response, it is called a Bode plot. A Bode plot is an ideal plot of filter frequency response because it assumes that the voltage gain remains constant in the passband until the cutoff frequency is reached, and then drops in a straight line. The filter network voltage gain in dB is calculated from the actual voltage gain (A) using the equation AdB = 20 log A where A = Vo/Vin. An ideal filter has an instantaneous roll-off at the cutoff frequency (fc), with full signal level on one side of the cutoff frequency. Although the ideal is not achievable, actual filters roll-off at -20 dB/decade or higher depending on the type of filter. The -20 dB/decade roll-off is obtained with a one-pole filter (one R-C circuit). A two-pole filter has
- 5. two R-C circuits tuned to the same cutoff frequency and rolls off at -40 dB/decade. Each additional pole (R-C circuit) will cause the filter to roll off an additional -20 dB/decade. In a one-pole filter, the phase between the input and the output will change by 90 degrees over the frequency range and be 45 degrees at the cutoff frequency. In a two-pole filter, the phase will change by 180 degrees over the frequency range and be 90 degrees at the cutoff frequency. Three basic types of response characteristics that can be realized with most active filters are Butterworth, Chebyshev, and Bessel, depending on the selection of certain filter component values. The Butterworth filter provides a flat amplitude response in the passband and a roll-off of -20 dB/decade/pole with a nonlinear phase response. Because of the nonlinear phase response, a pulse waveshape applied to the input of a Butterworth filter will have an overshoot on the output. Filters with a Butterworth response are normally used in applications where all frequencies in the passband must have the same gain. The Chebyshev filter provides a ripple amplitude response in the passband and a roll- off better than -20 dB/decade/pole with a less linear phase response than the Butterworth filter. Filters with a Chebyshev response are most useful when a rapid roll-off is required. The Bessel filter provides a flat amplitude response in the passband and a roll-off of less than -20 dB/decade/pole with a linear phase response. Because of its linear phase response, the Bessel filter produces almost no overshoot on the output with a pulse input. For this reason, filters with a Bessel response are the most effective for filtering pulse waveforms without distorting the waveshape. Because of its maximally flat response in the passband, the Butterworth filter is the most widely used active filter. A second-order (two-pole) active low-pass Butterworth filter is shown in Figure 3-1. Because it is a two-pole (two R-C circuits) low-pass filter, the output will roll-off -40 dB/decade at frequencies above the cutoff frequency. A second-order (two-pole) active high-pass Butterworth filter is shown in Figure 3-2. Because it is a two-pole (two R-C circuits) high-pass filter, the output will roll-off -40 dB/decade at frequencies below the cutoff frequency. These two-pole Sallen-Key Butterworth filters require a passband voltage gain of 1.586 to produce the Butterworth response. Therefore, and At the cutoff frequency of both filters, the capacitive reactance of each capacitor (C) is equal to the resistance of each resistor (R), causing the output voltage to be 0.707 times
- 6. the input voltage (-3 dB). The expected cutoff frequency (fc), based on the circuit component values, can be calculated from wherein, FIGURE 3 – 1 Second-order (2-pole) Sallen-Key Low-Pass Butterworth Filter FIGURE 3 – 2 Second-order (2-pole) Sallen-Key High-Pass Butterworth Filter
- 7. PROCEDURE Low-Pass Active Filter Step 1 Open circuit file FIG 3-1. Make sure that the following Bode plotter settings are selected: Magnitude, Vertical (Log, F = 10dB, I = -40dB), Horizontal (Log, F = 100 kHz, I = 100 Hz). Step 2 Run the simulation. Notice that the voltage gain has been plotted between the frequencies of 100 Hz and 100 kHz by the Bode plotter. Draw the curve plot in the space provided. Next, move the cursor to the flat part of the curve at a frequency of approximately 100 Hz and measure the voltage gain in dB. Record the dB gain on the curve plot. dB gain = 4.006 dB Question: Is the frequency response curve that of a low-pass filter? Explain why. Step 3 Calculate the actual voltage gain (A) from the measured dB gain. A = 1.586 Step 4 Based on the circuit component values in Figure 3-1, calculate the expected voltage gain (A) on the flat part of the curve for the low-pass Butterworth filter. Av = 1.586 Question: How did the measured voltage gain in Step 3 compared with the calculated voltage gain in Step 4? This result tells that we measured the exact voltage gain.
- 8. Step 5 Move the cursor as close as possible to a point on the curve that is 3dB down from the dB gain at the low frequencies. Record the dB gain and the frequency (cutoff frequency, fc) on the curve plot. dB gain = 0.965 dB fc = 5.321 kHz Step 6 Calculate the expected cutoff frequency (fc) based on the circuit component values. fc = 5305.16477 Hz Question: How did the calculated value for the cutoff frequency compare with the measured value recorded on the curve plot for the two-pole low-pass active filter? % difference = 0.298 % The measured value of cutoff frequency is not exactly the same with the computed value but the difference is too small. Step 7 Move the cursor to a point on the curve where the frequency is as close as possible to ten times fc. Record the dB gain and frequency (fc) on the curve plot. dB gain = -36.202 dB fc = 53.214 kHz Questions: Approximately how much did the dB gain decrease for a one-decade increase in frequency? Was this what you expected for a two-pole filter? The dB gain decreased by -37.167 dB which is approximately -40 dB. This is expected for a two-pole filter because it has two R-C circuits. Step 8 Click Phase on the Bode plotter to plot the phase curve. Change the vertical axis initial value (I) to 180 degrees and the final value (F) to 0 degree. Run the simulation again. You are looking at the phase difference (θ) between the filter input and output waveshapes as a function of frequency (f). Draw the curve plot in the space provided.
- 9. Step 9 Move the cursor as close as possible on the curve to the cutoff frequency (fc). Record the frequency (fc) and phase (θ) on the curve. fc = 5.321 kHz θ = -91.293 deg Question: Was the phase shift between input and output at the cutoff frequency what you expected for a two-pole low-pass filter? The phase will be 90 degrees at the cutoff frequency, so yes, it is what I expected. Step 10 Click Magnitude on the plotter. Change R to 1 kΩ in both places and C to 1 pF in both places. Adjust the horizontal final frequency (F) on the Bode plotter to 20 MHz. Run the simulation. Measure the cutoff frequency (fc) and record your answer. fc = 631.367 kHz Step 11 Based on the new values for resistor R and capacitor C, calculate the new cutoff frequency (fc). fc = 159.1549 MHz Question: Explain why there was such a large difference between the calculated and the measured values of the cutoff frequency when R = 1kΩ and C = 1pF. Hint: The value of the unity-gain bandwidth, funity, for the 741 op-amp is approximately 1 MHz. There was such a large difference between the calculated and measured values of the cutoff frequency because the calculated fc exceeds the funity of the op-amp
- 10. High-Pass Active Filter Step 12 Open circuit file FIG 3-2. Make sure that the following Bode plotter settings are selected: Magnitude, Vertical (Log, F = 10dB, I = -40dB), Horizontal (Log, F = 100 kHz, I = 100 Hz). Step 13 Run the simulation. Notice that the voltage gain has been plotted between the frequencies of 100 Hz and 100 kHz by the Bode plotter. Draw the curve plot in the space provided. Next, move the cursor to the flat part of the curve at a frequency of approximately 100 kHz and measure the voltage gain in dB. Record the dB gain on the curve plot. dB gain = 3.776 dB Question: Is the frequency response curve that of a low-pass filter? Explain why. Yes, Step 14 Calculate the actual voltage gain (A) from the measured dB gain. A = 1.5445 Step 15 Based on the circuit component values in Figure 3-2, calculate the expected voltage gain (A) on the flat part of the curve for the high -pass Butterworth filter. Av = 1.586 Question: How did the measured voltage gain in Step 14 compare with the calculated voltage gain in Step 15? The measured voltage gain differs with the calculated voltage gain with a little percentage.
- 11. Step 16 Move the cursor as close as possible to a point on the curve that is 3dB down from the dB gain at the high frequencies. Record the dB gain and the frequency (cutoff frequency, fc) on the curve plot. dB gain = 0.741 dB fc = 5.156 kHz Step 17 Calculate the expected cutoff frequency (fc) based on the circuit component values. fc = fc = 5305.16477 Hz Question: How did the calculated value of the cutoff frequency compare with the measured value recorded on the curve plot for the two-pole low-pass active filter? The measured value and the calculated value had a percent difference of 2.89 % and yet still approximately the same. Step 18 Move the cursor to a point on the curve where the frequency is as close as possible to one-tenth fc. Record the dB gain and frequency (fc) on the curve plot. dB gain = -36.489 dB fc = 515.619 Hz Questions: Approximately how much did the dB gain decrease for a one-decade decrease in frequency? Was this what you expected for a two-pole filter? The dB gain decreased -37.23 dB which is approximately -40 dB. This is expected for a two-pole filter. Step 19 Change the horizontal axis final setting (F) to 50 MHz on the Bode plotter. Run the simulation. Draw the curve plot in the space provided.
- 12. Step 20 Measure the upper cutoff frequency (fc2) and record the value on the curve plot. fc2 = 24.779 kHz Question: Explain why the filter frequency response looked like a band-pass response when frequencies above 1 MHz were plotted. Hint: The value of the unity- gain bandwidth, funity, for the 741 op-amp is approximately 1 MHz.
- 13. CONCLUSION: I, therefore conclude that active filters use active devices such as op-amps combined with passive elements. Active filters have several advantages over passive filters. The passive elements provide frequency selectivity and the active devices provide voltage gain, high input impedance, and low output impedance. The passband voltage gain should be 1.586 to produce the Butterworth response. When the resistance of the resistor R and the capacitance of capacitor C is decreased, the cutoff frequency will increase a lot.