Lab 9.docx

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Jan 9, 2024

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ECEN 214 - Lab Report Lab Number: 9 Lab Title: Sinusoidal Steady State Response of a 2nd Order Circuit

Purpose/Goals: The main purpose of this lab is to help us understand how circuits respond to sinusoidal input voltages. This is done by replicating the circuit from Lab 8, but replacing the input voltage with a sinusoidal (AC) voltage. The resistors and capacitors are then switched to see how the response of the circuit changes. We also worked on how to use complex phasors to a practical sinusoidal signal. ● Understand the response of circuits due to a sinusoidal input. Procedure: This lab is broken up into two tasks. The first part is to build the Sallen-Key which can be seen in Figure 1. Figure 1: Sallen-Key Order Circuit (the image on the right is the circuit diagram, while the 2 ?? image on the left is the diagram built in the lab) The values of the components should produce a Q-factor of 1.5 and frequency of 400 Hz. Using the wave generator, generate a sine wave for the input voltage and record the input and output voltages of the oscillator. Be sure to measure the amplitude of the input and output voltage as well as the phase difference between the input and output voltage. Measure and record the input and output voltage of the circuit while varying the input frequency for the circuit. The

next step for this task is to adjust the frequency of the input sine wave until the amplitude of the output to the input amplitude has a ratio of 0.707. This is referred to as the cut-off frequency so be sure to record this value as well as the phase difference at this frequency. The next part of this lab is to repeat the last task, but using the modified circuit in Figure 2 to see how the placement of the capacitors and resistors will affect the output and input voltages. Figure 2: Modified Sallen-Key Order Circuit (the image on the right is the circuit diagram, 2 ?? while the image on the left was the circuit built in the lab) Results: For Task 1 and Task 2, we used the values = 1500 Ohms, = 5100 Ohms, = 470 ? 1 ? 2 𝐶 1 nF, and = 39 nF. These values were found by plugging them into Equations 1, 2, and 3. This 𝐶 2 gave us a closest approximation for the Q-factor of 1.5 and radian frequency of 800π. Data Tables: By analyzing the data in Table 1, we see that as the input frequency increases, the amplitude of output voltage decreases. At 1000Hz, the amplitude of output voltage basically becomes zero. It is important to note that at the input frequency of 316Hz, the amplitude of output voltage unreasonably increases. This could be due to improper measurements by either the machine or human error. Based on the data in Table 1, it can be concluded that the circuit in Figure 1 is a low-pass filter. The cutoff frequency for the Figure 1 circuit is found to be 590Hz, which agrees with our data because as input frequency increases the amplitude of output voltage should decrease. This leads to a lower ratio in the amplitude of output voltage to input voltage.

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By analyzing the data in Table 2, we see that as the input frequency increases, the amplitude of output voltage increases. This is the opposite trend we saw in Table 1. Therefore, it can be reasonably assumed that the circuit in Figure 2 is a high-pass filter. This idea is supported as the amplitude of output voltage is relatively similar to the input voltage at higher frequencies. At low frequencies, for example at a frequency of 10Hz, the amplitude of output voltage is practically zero. It is found that the cutoff frequency for Figure 2 is 620Hz, which is supported by the data. The cutoff frequency equation is represented by Equation 4. Input freq (Hz) Amplitude input (V) Amplitude output (V) Phase difference (degrees) 10 2.21 2.21 3.13 18 2.17 2.21 2.82 32 2.21 2.21 5.50 56 2.21 2.25 5.4 100 2.17 2.33 11.88 178 2.21 2.53 19.1 316 2.37 3.54 59.19 562 2.17 1.57 138.94 590 2.13 1.53 142.4 1000 2.13 600 mV 175.37 1778 2.13 400 mV -747.0 3162 2.13 320 mV -187.3 5600 2.13 320 mV -409.3 10000 2.05 280 mV 166.62 Table 1: Values measured for the Sallen-Key Order Circuit as the input frequency is 2 ?? increased to 10000Hz.

Input freq (Hz) Amplitude input (V) Amplitude output (V) Phase difference (degrees) 10 2.21 320 mV -72.70 18 2.29 280 mV 119.10 32 2.25 320 mV -8.77 56 2.25 360 mV 179.39 100 2.29 520 mV -61.4 178 2.21 560 mV 231.1 316 2.21 960 mV -100.91 562 2.17 1.49 -70.4 620 2.17 1.51 1.57 1000 2.25 1.89 -44.09 1778 2.21 2.13 -23.6 3162 2.29 2.25 -13.4 5600 2.25 2.17 -4.41 10000 2.29 2.21 -0.1 Table 2: Values measured for the Modified Sallen-Key Order Circuit as the input frequency 2 ?? is increased to 10000Hz. Data Plots:

The frequency axis should be presented on a logarithmic scale because the relationship between the input frequency and the amplitude of output voltage is not linear. From the Tables, it can be seen that the amplitude of output voltage severely changes as input frequency is altered. In Figure 3, we can see that it is a low pass filter because as frequency increases the amplitude of output voltage decreases to nearly 0. In Figure 4, the phase difference after 1000Hz should be ignored because the low pass filter doesn’t allow high frequencies to pass through resulting in the off data after that point. In Figure 5, we can see that it is a high pass filter because as frequency increases the amplitude of voltage output increases and at low frequency it is near 0. Again, because it is a high pass filter the lower frequencies before 620Hz can be ignored as low frequency won’t pass through resulting in off numbers. Figure 3: Amplitude of Output Voltage of Figure 1 as frequency is increased. Figure 4: Phase Difference of Figure 1 as frequency is increased.

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Figure 5: Amplitude of Output Voltage of Figure 2 as frequency is increased. Figure 6: Phase Difference of Figure 1 as frequency is increased. Calculations: In order to calculate the values that will create Q-factor of 1.5 and 𝐶 1 , 𝐶 2 , ? 1 𝑎?? ? 2 resonant radian frequency of = 800π, Equations 1, 2 and 3 are used. ω 0 Equation 1 ? ?𝑞 = ? 1 ? 2 ? 1 +? 2 Equation 1 is used to find the value of from resistor values in order to be ? ?𝑞 ? 1 𝑎?? ? 2 plugged into Equation 2. Equation 2 ? = 𝐶 1 𝐶 2 ? ?𝑞 ? 1 +? 2 Equation 2 is used to relate the capacitor and resistor values used in the circuit to the Q-factor. By plugging in certain resistor and capacitor values, one can calculate the Q-factor of the circuit. In this case, it is primarily used to find which set of resistor and capacitor values will result in a Q-factor of 1.5.

Equation 3 ω 0 = 1 ? 1 𝐶 1 ? 2 𝐶 2 Equation 3 is used to relate the capacitor and resistor values used in the circuit to the resonant radian frequency. In this lab, we are looking for resistor and capacitor values that will result in a resonant radian frequency of 800π. By plugging in random values into this equation, we can find a proper set of resistors and capacitors that will result in 800π. Equation 4 |𝑉 ??? (?)| |𝑉 𝑖? (?)| = 1 2 = 0. 707 The purpose of Equation 4 is to find the cut-off frequency of the circuit, which allows us to understand when the output voltage will fall below 70.7% of the input voltage. Discussion: For the circuit in Figure 1, the range of frequencies that pass through the circuit relatively unaltered are from 10Hz to 590Hz. The frequencies that are highly attenuated are from 1000Hz to 10000Hz. For the circuit in Figure 2, the range of frequencies that pass through the circuit relatively unaltered are from 562Hz to 10000Hz. The frequencies that are highly attenuated are from 10Hz to 316Hz. To adjust the ranges of the pass bands and stop bands, one can decrease the values of in Figure 1 to allow higher frequencies to pass through the circuit 𝐶 1 , 𝐶 2 , ? 1 𝑎?? ? 2 without affecting the amplitude of the output voltage much compared to the amplitude of the input voltage. To be more precise in finding the frequencies where it can pass through the circuit relatively unaltered or highly attenuated, we would increase the input frequency in smaller intervals. Conclusion: In this lab, we built a Sallen-Key Order circuit to help us understand how circuits 2 ?? respond to sinusoidal input voltages. From the data collected in Table 1, it can be inferred that by creating the circuit in Figure 1, we have created a low-pass filter. The data supports this idea because as the input frequency increases, the amplitude of the output voltage decreases. Around the input frequency of 1000Hz, the amplitude of the output voltage is extremely close to 0. However by creating the circuit in Figure 2, the circuit suddenly becomes a high-pass filter. The data in Table 2 supports this because when the input frequency is extremely high, the output amplitude voltage is roughly equivalent to the input amplitude voltage. For a low input frequency, the output amplitude voltage is close to 0, which is significantly lower than the input amplitude voltage. Overall, all lab goals were accomplished as we were able to further understand the response of circuits with a sinusoidal input.

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