One way to obtain second order systems is to combine two first order systems. Two fundamental ways of combining two or more systems are the series (or cascade) and parallel connections. In this lab you are going to combine a first order lowpass filter (LPF) and a first order highpass filter (HPF) to design a bandpass filter (BPF) and a bandstop filter (BSF).
A first order LPF with input phasor VS1 and output phasor VO1 has transfer function (or phasor gain)
where w1 is the cutoff frequency (or half-power frequency of -3dB frequency). The magnitude and the phase of H1 versus w are plotted in the following two graphs.
An implementation of the transfer function H1 is the RC LPF shown below with w1 = (R1C1)-1.
A first order HPF with input phasor VS2 and output phasor VO2 has transfer function (or phasor gain)
where w2 is the cutoff frequency (or half-power frequency of -3dB frequency). The magnitude and the phase of H2 versus w are plotted in the following two graphs.
An implementation of the transfer function H2 is the RC LPF shown below with w2 = (R2C2)-1.
A BPF can be obtained by cascading the transfer functions of a LPF and a HPF, with suitably chosen w1 and w2, as shown in the following block diagram.
A possible implementation of this using RC circuits with R1C1 = (w1)-1, R2C2 = (w2)-1, and an OpAmp is shown in the schematic below:
Using a LPF and a HPF in parallel rather than in series, as shown in the block diagram below, results in a BSF for suitably chosen w1 and w2.
E1. Design of BPF. Draw the following circuit in PSpice.
The design goal is to select R1, C1, R2, and C2, so that the circuit acts as a bandpass filter with center frequency fc = 600 Hz and as narrow a -3dB bandwidth as possible. Use standard capacitor values in the range of about 47 nF to 470 nF. The following figure shows the frequency response magnitude graph of a typical BPF.
Note that the -3dB bandwidth is measured relative to the maximum gain at or near fc. Note also that the lower and upper -3dB frequencies are not necessarily symmetric around fc.
E2. Building and Testing of BPF. On your breadboard, build and test the BPF that you designed in E1. Use the oscilloscope to measure the center frequency fc and the -3 dB frequencies. What is the bandwidth of this BPF? Do the measured results agree with the PSpice simulations? What is the step response of the BPF? Does it have the shape that you expected?
E3. Using Soundcard and Matlab for Measurements. In the future it will be convenient to reduce the time that it takes to make step and frequency response measurements and plot them. This can be achieved by capturing the input and output signals of circuits using the sound card of the computer, followed by processing and plotting the results in Matlab.
The sound card probe with amplifier which is used to pick up the signals to be measured is shown below. Plug the 1/8" mini jack connector into the line input (blue) at the back of the computer. Use the ±15 V power supply to power the probe. The green jumper lead goes to the ground of the circuit to be measured. The white jumper lead (left channel) is used to measure the input to the circuit and the red jumper lead (right channel) is used to measure the response from the circuit. The switches can be used to select either ×1 or ×10 amplification. For most measurements the ×1 position should work fine.
The next figure shows the inside of the sound card probe with amplifier.
The schematic of the sound card probe is shown below.
To record the step response of a circuit, generate a rectangular signal of about 0.5 Vpp with the waveform generator and use use it as input. If necessary, select the line input and adjust the level using the Volume Control panel shown below. To bring up the volume control right-click on the speaker symbol at the bottom right of the computer screen and select "Open Volume Controls". As an aternative you can also run "sndvol32.exe" under "Run" in the start menu. By default you will get the Playback Control panel. Click on "Options" and "Properties" and then select the "Recording" radio button.
To begin recording, launch Audacity 1.2.6 (click here to download). Make sure Line-in is selected, the recording device is the sound card, 2 channels (stereo) is selected, the sampling rate is 44100 Hz, and the sample format is 16 bits. More details on setting up and using Audacity 1.2.6 can be found here. Then click on the record button and record about 3 sec of the step response. The figure below shows an enlarged portion of such a recording. Check that the input level is large enough to be clearly visible, but not distorted. Save the recording as a wav-file for later processing in Matlab.
To record the frequency response of a system, generate a sweeping sine signal from about 50 Hz to 10000 Hz using the waveform generator in sweep mode. Use a sweep time of about 5 to 10 sec for the whole range. A typical recording (but with shortened sweep time to obtain a nicer picture) is shown below.
Select one sweep using the cursor and the export the selection as a wav-file for later processing in Matlab.
The following Matalb script file can be used to read and display wav-files made with the sound card probe and recorded and trimmed in Audacity 1.2.6.
Using this with the step response wav-file recorded earlier yields the following graph.
To see more details and to measure some features of the step response, use the zoom function of the Matlab figure and select a small portion of the time axis to zoom into. A plot that was obtained in this way is shown below.
Record the step response and the frequency response (in the range 50 Hz to 10000 Hz) of the BPF that you built in E2. Plot the wav-files in Matlab and use the zoom function in the Matlab figures to determine the time constant of the step response and the center frequency and the -3dB frequencies of the frequency response.
E4. Design and Measurement of BSF. Design a bandstop filter with center frequency fc = 600 Hz, using a first order RC LPF and a first order RC HPF in the following block diagram.
The goal is to attenuate fc by 20 dB (corresponding to an amplitude attenuation by a factor of 10), while keeping the -3dB bandwidth as narrow as possible.
Start from designing and simulating the circuit in PSpice. Once you have a suitable design, build it on your breadboard and measure its frequency response and its step response. You may either use the oscilloscope or the sound card probes for these measurements.
Lab worksheet in PDF format: lws02.pdf
Note: Each student needs to turn in a lab worksheet. The raw measured data for each student in a lab team will be the same, but the conclusions drawn from it are individual to each partner in the team.
©2003-2007, P. Mathys. Last revised: 1-30-07, PM.