Lab 8.docx

pdf

School

Texas A&M University *

*We aren’t endorsed by this school

Course

214

Subject

Electrical Engineering

Date

Jan 9, 2024

Type

pdf

Pages

Uploaded by MateKnowledgeMonkey156

ECEN 214 - Lab Report Lab Number: 8 Lab Title: Transient Response of a 2nd Order Circuit

Introduction and Goals: The main objective of this lab is to learn about a 2nd order response with the use of an op-amp circuit. In this lab, we are primarily going to be using resistors and capacitors to produce the 2nd order response. By using certain resistor and capacitor values, we can dictate the type of response the order circuit will have. At a quality factor of 0.5, the system will produce a 2 ?? critically damped response. At a greater quality factor the system will be underdamped, while a quality factor below 0.5 will result in an overdamped response. Using this knowledge, we will compare the peak-to-peak voltage to the input voltage of the circuit. The type of circuit that will be used in this lab is called a Sallen-Key Order circuit. 2 ?? ● Gain a better understanding of op-amp circuits that create a order response. 2 ?? ● Understand how the step response of the Sallen-Key Order circuit measured in the 2 ?? lab differs from theory and simulation results. Procedure: For this lab, the Sallen-Key Order circuit is created (shown in Figure 1). Using 2 ?? Equations 1, 2, and 3, the resistor and capacitor values that produce a critically damped response are found (Q = 0.5). Input the found resistors and capacitors into the circuit shown 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 circuit built in the lab) The wave generator is then used to produce a 100 Hz square wave with an input voltage (peak-to-peak) of 2 volts. Display the input wave on CH1 and output wave on CH2 using the oscilloscope, making sure to record the output waves frequency and peak-to-peak voltage. Repeat this process with different Q-factors that show overdamped and under damped circuits. This can be accomplished by changing the resistor and capacitor values.

Data Tables: From Table 2, it can be interpreted that at an underdamped response, the peak-to-peak output voltage is greater than the peak-to-peak input voltage. For an overdamped response, the peak-to-peak output voltage is less than the peak-to-peak input voltage. At a critically damped response, the input and output peak-to-peak voltage are equal. Q-factor (Ω) ? 1 (Ω) ? 2 (nF) ? 1 (nF) ? 2 0.5 390 390 100 100 0.26 6100 500 100 100 0.13 6100 100 100 100 2.57 5100 100000 20 0.5 1.022 500 1000 470 100 Table 1: Resistor and capacitor values used to create an overdamped, underdamped and critically damped response in the circuit. The Q-factor of 0.5 is especially important, as it is the fine line between an underdamped and overdamped response. Q-factor Output (V) 𝑉 ?? Frequency (Hz) 2.57 4.12 100.01 1.022 2.85 100.02 0.13 2.09 100.05 0.26 2.11 100 0.5 2.11 100 Table 2: Measured peak-to-peak output voltage at a given Q-factor. It can be seen that at an underdamped response, the peak-to-peak output voltage is greater than both the critically damped and overdamped response. Data Images: From the figures below, it can be interpreted that at an underdamped response, the peak-to-peak output voltage is greater than the peak-to-peak input voltage. However, for an overdamped response it is expected for the peak-to-peak output voltage to be less than the input. This only happened in Figure 4, but for Figure 5 the increase in voltage could be due to

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

measurement error. The same error could be seen in Figure 6 for a critically damped response, the input and output peak-to-peak voltage should be equal, but the output was slightly greater. Figure 2: Input and output waveforms at a Q-factor 2.57. It can be seen that the output peak-to-peak voltage is greater than the input. The yellow waveform is the input, while the output waveform is green.`

Figure 3: Input and output waveforms at a Q-factor 1.022. It can be seen that the output peak-to-peak voltage is greater than the input. The yellow waveform is the input, while the output waveform is green.`

Figure 4: Input and output waveforms at a Q-factor 0.13. It can be seen that the output peak-to-peak voltage is less than the input. The yellow waveform is the input, while the output waveform is green.` Figure 5: Input and output waveforms at a Q-factor 0.26. It can be seen that the output peak-to-peak voltage is slightly greater than the input. Given the current trends, the output should be slightly less. The yellow waveform is the input, while the output waveform is green.`

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Figure 6: Input and output waveforms at a Q-factor 0.5. It can be seen that the output peak-to-peak voltage is slightly greater than the input. Given the current trends, the output should be the exact same. The yellow waveform is the input, while the output waveform is green.` Results: (1) ? = ? 1 ? 2 ? ?? ? 1 +? 2 We are able to calculate the Q-factor to measure how damped the 2nd order circuit is with Equation 1. We are also able to use the values of the resistor and capacitors for the other values that are needed for the differential equations. (2) ? ?? = ? 1 ? 2 ? 1 +? 2 (3) ω 0 = 1 ? 1 ? 1 ? 2 ? 2 (4) ω ? = 1 ? 1 ? 1 ? 2 ? 2 − ( 1 2? ?? ? 1 ) 2 (5) α = 1 2? ?? ? 1

With these values, we are also able to find other values for the differential equation for the 2nd order circuit. The differential equation that is used is for an underdamped equation is: (6) 𝑉 ??? (?) = ? 1 ? −α? ???(ω ? ?) + ? 2 ? −α? ?𝑖?(ω ? ?) For an overdamped, the equation is: (7) 𝑉 ??? (?) = ? 1 ? ? 1 ? + ? 2 ? ? 2 ? For a critically damped, the equation is: (8) 𝑉 ??? (?) = ? 1 ?? −α? + ? 2 ? −α? For the s-values, would use the equation: (9) ? = −(? 1 +? 2 )? 2 ± (? 1 +? 2 )? 2 2 −4? 1 ? 1 ? 2 ? 2 2? 1 ? 1 ? 2 ? 2 Q = 0.5 R1 = 390 Ohms R2 = 390 Ohms C1 = 100 nF C2 = 100 nF LTSpice Simulations: Figure 7: Square wave graph for Q= 0.544. Compared to Figure 6, the output waves look very similar to the input waves. This should be the case as the circuit is critically damped at this Q-factor.

Figure 8: Square Wave graph for Q = 0.287 which does match the output of Figure 5. It is very similar in shape and supports the idea that the output voltage should resemble closely to the input voltage. Figure 9: Square Wave graph for Q = 0.0875. Compared to the input wave of Figure 3 it is similar.

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Figure 10: Square wave graph for Q = 1.022. Compared to waveforms in Figure 3, the waveforms don’t look alike. This could be due to noise from the circuit built in the real-world. Figure 11: Square Wave graph for Q = 2.53. Compared to Figure 2, the waveforms don’t look anything alike. This could be caused by noise the circuit would experience in real-life or incorrect measurements. Discussion: In this lab, we learned how certain resistor and capacitor values cause an underdamped, overdamped or critically damped response from a second order circuit. From our data, it can be seen that as the Q-factor decreases below 0.5, the output peak-to-peak voltage should decrease below the input voltage. As the Q-factor increases, the output peak-to-peak voltage should increase above the input voltage. The most likely causes for the difference in theoretical and the

real circuit could be due to resistors and capacitors not being the exact value (percent error range) or the oscilloscope not measuring as precisely as possible due to internal resistance. Conclusion: The main objective of the lab was to further understand how a 2nd order circuit works and its responses to different resistance and capacitor values. Another objective was to compare how the theory behind the Sallen-Key 2nd order circuit compares to the measurements found in the lab. Both objectives were accomplished as we can see from the LTSpice results that some of the waveforms did resemble the output waveforms we saw in the lab. However, the shape of the theoretical circuit didn’t exactly match the one we saw in the oscilloscope. We also learned that depending on if it is critically damped, overdamped or underdamped, it affects the output voltage of the waveform. Signatures: