This laboratory experiment has the following tasks:
1. Build and take quantitative measurements of the Sallen-Key band pass audio filter.
2. Perform a qualitative assessment of the filter by first playing and audio recording through the filter, and then without the filter while connected to a speaker.
When an audio signal is sent through a phone, it sounds different than the original source of sound. There are many factors that alter the original signal such as the components of the telephone system. The signal can lose a lot of high and low frequency information through the microphone and speakers for example. Some amount of this information is also lost intentionally. The intentional loss of these frequencies is used through a frequency division multiplexing system to use modulation to get several phone calls onto one cable. To do this, a band pass audio filter is used to restrict the frequencies and thus the bandwidth of the phone calls. This experiment details the analysis of a Sallen-Key band pass audio filter and will evaluate the sensitivity of the components to determine if it is plausible to use cheaper 20% tolerance components instead of higher priced lower-tolerance components.
The Sallen-Key filter is an electronic filter that can be configured to implement high-pass, low-pass, bandpass, and band reject filters. This experiment will focus on the bandpass configuration. This configuration has a transfer function that can be found by calculating the voltage gain of the op amp. This is found with the gain formula V0/Vs where V0 is the output signal and Vs is the input signal. To find this, a few characteristics of the circuit must be taken into account.
· There is no current flowing into the positive terminal of the op-amp.
· Both the positive and negative terminal follow the virtual short property, they have the same voltage.
· Because of the lack of current flowing into the positive terminal of the op-amp, a relation can be found using voltage division using the nodes to the left and right of C2.
Writing KCL equations for the nodes and relating all of the equations together reveals the full equation for the gain of the Sallen-Key Band Pass Filter. This equation represents the transfer function which was then analyzed for various values of C1 ranging from 0.1nF to 1μF. Using the above formula to calculate the gain and the formula for the center frequency shown below, data was collected for the center frequency and the 3dB points. The mean and standard deviation of these values is also shown in the table below.
As shown in the table, while the gain stays the same, the 3dB points and center frequency decreases as the value of C1 is increased. The data collected uses values of C1 that are very spaced apart so the values of the mean and standard deviation may be misleading and don’t give any insight on how statistically relevant the information is. Had the experiment been conducted a second time, the values would have been calculated based on varying C1 on the same 20% tolerance interval used throughout this lab. This data will be referenced in the later analysis of the simulation of the transfer function and further comparison to the results of the lab.
After the circuit was constructed, the values for C1 were varied by using the capacitance substitution box. The capacitance for C1 was to be varied within a 20% tolerance of 47nF so the values for C1 were varied within the interval of [37.6nF, 56.4nF]. These values were set while keeping the input level low so that the signal did not clip. This was done by lowering the output on the function generator. The values for f0, Hmax, and B were recorded and are shown in the table.
The values for the max amplitude and bandwidth were found by measuring with the cursers using the scopy oscilloscope software. The bandwidth was calculated by subtracting the 3dB frequencies which were found at the frequency corresponding with 0.707 * Hmax. As shown in the table, as the value of C1 is increased, f0 decreases and Hmax also decreases. Bandwidth stays relatively the same.
After these values were collected, measurements were then taken for the gain using the provided oscilloscope though the scopy software. This was done to measure the gain (Vout/Vin) as a function of C1 for three different values of C1.
To measure these values, C1 was set at 47nF, 37.6nF, and 56.4nF and the nodes corresponding to Vin and Vout of the op amp were measured for frequencies ranging from 20Hz to 20000Hz. As shown in the table of figure 6, the gain reaches a peak for all configurations of the circuit at about 450Hz before decreasing as the frequency goes to 20kHz.
The second part of the experiment involved a qualitative analysis of the filter by running a voice recording through the filter. The output of the filter was connected to a small speaker and a probe providing the signal from the audio recording was sent through the input. The voice recording was loaded into the signal generator as a wav file and was then played through the speaker. This was done for the nominal and the 20% tolerance points. Then the audio was played through the speaker without the filter. Detailed analysis of the quality of the audio is included in the qualitative analysis section.
As shown in the graph for the gain of the measured transfer function, when C1 is 20% smaller than its nominal value, the gain is overall larger. When C1 is 20% greater than its nominal value, the gain is smaller. This follows the general consensus so far that greater values of capacitance correspond with smaller values of frequency.
Measured Hmax of transfer function
Measured frequency of transfer function
The graphs for both Hmax and the frequency measured for the transfer function show a decrease as the capacitance increases. This is expected, as the 3dB frequency was found to decrease as the capacitance increases and the 3dB frequency is dependent on the frequency. Both graphs show a relatively linear decreasing plot which is not the case for the bandwidth. The bandwidth stays relatively the same regardless of a change in capacitance or frequency. In the context of an audio filter this makes sense as the bandwidth would need to stay consistent to ensure that each call has the same amount of space in the frequency spectrum.