|Brandeis University||Physics 29a|
|Spring 2018||Kevan Hashemi|
The circuit below is a second-order, single-stage active filter. The ratio R1/(R1+R2) is the nominal gain of the filter. The product RC is its time constant. Look at the Active Filter sheet of our Filter Design Guide spreadsheet, which you will find on the 29A page. You will see example values for the gain and time constant of low-pass filters with cut-off frequency 1 rad/s. The time constant of a filter with cut-off frequency ω rad/s will be ω times smaller than for 1 rad/s.
We make a high-pass activer filter by swapping the resistors of value R in the above circuit for the capacitors of valuce C. To make a fourth-order active filter, we connect two of the above circuits in series. The pass-band of a filter is the range of frequencies we expect the filter to pass from input to output. Outside the pass-band, we expect the filter to stop signals from reaching the output. The pass-band of a low-pass filter is the range zero up to the cut-off frequency. Whenever we design a filter, we trade sharpness of the cut-off response for uniformity of pass-band response. The Butterworth filter provides uniform gain in the pass-band, but its cut-off is not as sharp as the 3-dB Chebyshev filter, which allows 30% gain variation in the pass band. The 0.5-dB Chebyshev filter is a compromise: allows 10% gain variation in the pass-band.
Part 1: Make a second-order, active low-pass filter with cut-off frequency close to 1 kHz. Use the TL081 op-amp. Choose whatever filter response you like. Use the Filter Design Guide spreadsheet to plot the theoretical response of the filter. Measure its gain versus frequency at ten or more points and compare your measurements to your prediction.
Part 2: Build two active-filter circuits and connect them in series to make a four-pole, 3-dB Chebyshev low-pass filter with cut-off frequency close to 1 kHz. Measure its frequency response and compare to that of your second-order filter.
Part 3: Convert your fourth-order low-pass filter into a high-pass filter by swapping the appropriate resistors and capacitors. Measure the gain versus frequency. Is the new cut-off frequency the same as the old?