EME108S24_Lab3_Handout

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Apr 25, 2024

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1 University of California Davis EME 108 Spring 2024 Lab 3: OPERATIONAL AMPLIFIERS Learning Objectives Equipment and software 1. Learn about different types of op-amp circuits and their uses. 2. Gain practical experience in assembling circuits and calculating output voltage in electric circuits. 3. Build an inverting circuit, a summing circuit, and an integrating circuit. 4. Learn how op-amps work and determine why filtering is important. 5. Construct and analyze an operational amplifier first order system. 1. Three 𝜇𝐴741 op-amps 2. Five 100𝑘𝛺 resistors 3. One 0.01𝜇𝐹 capacitor 4. One potentiometer ( 100𝑘𝛺 max) 5. One breadboard and wires for connections 6. MyDAQ function generator and oscilloscope 7. MyDAQ Bode tool A) Lab Assignments: 1. Before your lab section starts, read through the lab handout and complete the pre-lab work in your notebook, with the following entries all in your own words (refer to EME108 Lab Notebook Template.pdf on Canvas for more information on how to write the notebook reports). Every student needs to complete their own pre-lab entry in their own notebook and let the TAs check at the start of the lab. - Title block - Objective - Equipment and setup - Responses to pre-lab questions - Concise procedure in bullet points Every student needs to complete their own pre-lab entry in their own notebook and let the TAs check at the start of the lab ( first 15 minutes). 2. Lab memo report, to be submitted to Canvas before the start of your next lab session. Each group only needs to submit one report. - Review EME108_memo_report_guidelines.pdf and EME108S24_Lab3_Memo_Instructions.pdf and use them as a checklist. - Refer to documentations provided on Canvas in the “ Engineering Writing Resources ” folder to help creating your report. B) Scenario: Your boss suddenly becomes extremely interested in 1 st order systems and asks you to construct an electrical circuit, which operates as a 1 st order system, and then analyze it for a range of input frequencies and use different methods to determine its time constant and cut-off frequency.

2 C) Prelab Questions: Before your lab section, review how to calculate output voltage in electric circuits, as explained in class. Answer the following questions in your lab notebook: 1. Consider the composite circuit shown in Figure 5. This composite circuit is the union of the three individual circuits shown in Figures 2 to 4. Use the equations in Figures 2 to 4 to write a differential equation that relates the input voltage ( 𝑒 ௜ ) and output voltage ( 𝑒 ௢ ) of the circuit in Figure 5. For your calculations, assume 𝑅 ଵ = 𝑅 ଶ = 𝑅 ଷ = 𝑅 ସ = 𝑅 ହ . (Hint: the result will be a first-order differential equation. 𝑅 ௜ is the potentiometer resistance and is variable.) 2. Assume the input to the system in Figure 5 is a sinusoidal signal 𝑒 ௜ (𝑡) = 𝐴 ∗ 𝑠𝑖𝑛 (𝜔𝑡) , the steady-state output 𝑒 ௢ of this system will also be a sinusoid 𝑒 ௢ (𝑡) = 𝐵 ∗ 𝑠𝑖𝑛 (𝜔𝑡 + 𝜙) with a different amplitude 𝐵 and a different phase angle 𝜙 . Use the result from Question 1 and your knowledge of the first-order system, write the expressions for 𝐵 and 𝜙 . 3. Based on your results from Question 2, write the expression for the magnitude ratio (output amplitude divided by input amplitude) of the system in Figure 5 and provide a sketch of its plot versus frequency. 4. Based on your results from Question 2, write the expression for the phase shift of the system in Figure 5 and provide a sketch of its plot versus frequency. 5. Use your knowledge of the first-order system and your answers from Questions 1-4, write the expression for the time constant of the composite circuit in Figure 5. 6. If we have a fast-moving input signal that we want the system in Figure 5 to follow, would we want to have a time constant of the system that is small or large? Explain why. 7. If we have a slow-moving input signal that we want the system in Figure 5 to follow, but a fast-moving noise on the input that we want to filter out, would we want to have a time constant of the system that is small or large? Explain why. 8. What’s the difference between angular frequency 𝜔 and circular frequency 𝑓 ? What are their units respectively? How are they related? 9. What does it mean if the dynamic system you are studying has a steady state gain greater than 1? D) Reference: Take a look at the following reference to improve your understanding of op-amp based circuits: 1. http://www.electronics-tutorials.ws/category/opamp/

3 E) Lab Procedures: Part 1: Testing Individual Circuits 1. Look at the pinout diagram of the op-amp shown in Figure 1, sketch in your lab notebook the wiring diagrams for the inverter, the summer, and the integrator circuit shown in Figures 2 to 4, respectively. These sketches should show which pins of the op-amp are being used for each circuit. You should understand your wiring diagrams before you attempt to build the circuits. Check your diagrams with your TA and obtain his/her signature . 2. Before building your circuit, measure the value of each resistor and each capacitor you are using to make sure you have selected the right components. Note: The potentiometer (“pot”) has three pins: call them top, middle, and bottom. The top and bottom pins are collinear, and the resistance between these two collinear pins is a constant value around 100𝑘𝛺 . Turning the adjustment screw will simultaneously change the resistance between the top and middle pins and the resistance between the middle and bottom pins. That is, if the top-middle resistance increases by 100 Ω, the middle-bottom resistance will decrease by 100 Ω. 3. Disconnect power to your myDAQ (USB cable) at this step. Using your sketches as the guide, strategically build these three individual circuits in Figures 2 to 4 with three op-amps in different places on your breadboard. Make sure to employ good circuit building practices! The three pins of the potentiometer need to be placed in three different rows on the breadboard, but wire only the top and middle pins of the potentiometer to correct op-amp pin locations, leaving the bottom pin plugged in but unused. Do not cut or bend the pins of your potentiometer. Keep your circuit clean and wires short, use railings for shared power and ground, and use color- coded wires. 4. Do not power to your myDAQ yet. You are going to use the myDAQ function generator to generate the input signal and the myDAQ oscilloscope to view both the input and the output signals. Your oscilloscope should display both the input and output waveforms. Hint: use the AO 0 channel of myDAQ to generate the input signal to your circuit, use AI 0+ to read the input signal and AI 1+ to read the output signal of your circuit. Don’t forget to connect AI0- and AI1- to ground. 5. Now test the expected functionality of each circuit . Ask your TA to check the circuit you are going to test and obtain his/her signature, before plugging the myDAQ to the computer. Disconnect the op-amps not in use from the +/-15V power during tests and remember to unplug the USB cable of the myDAQ between tests. 1) Summing circuit: Generate a sine wave with 1 kHz frequency, 2 Vpp amplitude and 0 V offset as an input , wire both 𝑒 ௜ and 𝑒 ଶ to the input signal from the myDAQ function generator. Does the output combine the signals as indicated by the given formula? Show your TA your results and obtain his/her signature.

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4 2) Integrating circuit: Change the input to a square wave of 1 kHz frequency, 2 Vpp amplitude, 0 V offset and 50% duty cycle . The integrator should give an amplified saw tooth wave as an output. If you don’t see the saw tooth response, disable the oscilloscope AI 0 channel and adjust the resistance of the potentiometer until you see it. You will see clipping in the output – why? Explain this in your lab notebook. Show your TA your results and obtain his/her signature. 3) Inverting circuit: Generate a sine wave with 1 kHz frequency, 2 Vpp amplitude and 0 V offset as an input. The inverter output should multiply the input waveform by “-1”, which causes a 180 ° phase shift. Show your TA your results and obtain his/her signature. Part 2: First-Order Low-Pass Filter Circuit 1. In your lab notebook, sketch a wiring schematic of Figure 5 using the pinout diagrams of the op-amps. Have your TA check off your sketch and obtain his/her signature. 2. Wire your circuit using your sketch as a guide, with myDAQ not connected to the computer. Connect AO 0 to e i , and use AI 0+ to read the input e i , AI 1+ to read the ouput e o . Don’t forget to connect AI 0- and AI 1- to ground. Make sure to employ good circuit building practices. Have your circuit checked off by your TA and obtain his/her signature, before powering the myDAQ. 3. Use the myDAQ function generator to send a square wave (1100 Hz frequency, 1 Vpp amplitude, 0.5 V DC offset, 50% duty cycle) to the input e i of your circuit and observe the output e o on the myDAQ oscilloscope. Vary the potentiometer resistance, 𝑅 ௜ , until the 1 st order step response on the oscilloscope is fully developed. Your response should have a very similar shape to Figure 6. Verify your step response with your TA and obtain his/her signature. DO NOT adjust the potentiometer resistance 𝑹 𝒊 starting from this point to the end of the lab. 4. Unplug the myDAQ from the computer and take your potentiometer out from the breadboard to measure the resistance between the top and the middle pins. Record this potentiometer resistance, 𝑅 ௜ . Then plug the potentiometer back and power your circuit. Make sure the potentiometer pins are plugged back correctly so that the oscilloscope will display the same step response as before you take the measurement. Potentiometer Resistance, 𝑅 ௜ : 5. Use equations from the pre-lab questions to determine the theoretical time constant 𝜏 . To get more accurate results, use the measured capacitance 𝐶 of the capacitor you are using. Then use the definition of the cut-off frequency to determine the theoretical cut-off frequency 𝑓 ௖ in Hz. Hint: Cut-off frequency is the frequency corresponding to - 45° phase shift, and it is also the frequency at which the output power drops to 1/2 of the input power. The cut-off frequency in Hz is related to the reciprocal of the time constant, 1/𝜏 , by the relationship 𝑓 ௖ = 1/(2𝜋𝜏) .

5 Theoretical time-constant 𝜏 : Theoretical cut-off frequency 𝑓 ௖ (Hz): 6. In the myDAQ oscilloscope window, trigger the data acquisition on a rising edge with the acquisition mode set to run once to make the plot “freeze” on the oscilloscope display (change to correct settings in the aera marked in blue in the figure below ). Check “cursors on” and set both cursors C1 and C2 to the output channel CH 1 (AI 1) (make changes to the area marked in red in the figure below ), then use the cursors (dashed lines) to estimate the time constant 𝜏 directly from the step response plot, interpolating the data if you cannot get the exact result. (Hint: to measure the time constant 𝜏 , consider its definition in Figure 6.) Take a screenshot of your 1st order step response (the plot you used to estimate the time constant). You will need to include this screenshot in your memo report and mark on the plot (can place cursors) the time constant. Time Constant 𝜏 estimated: 7. Download the data that forms your 1 st order step response plot. Make sure that the saved data contains one fully developed rising response and one fully developed decreasing response in one full period. To save the data, use the “Stop” button to stop the oscilloscope, click the “Log” button (marked in orange in the figure above), and export the data to a .txt file. Ask your TA to check your data and help you open the data in Excel, then obtain his/her signature. You will use this data to perform a curve fit as another estimate of the time constant 𝜏 in your memo report. 8. Now experimentally determine the cut-off frequency 𝑓 ௖ of your circuit by comparing the phase shift between the input and the output of your circuit. Use a 1 Vpp sine wave as an input, start with an input frequency that is 100 Hz less than the theoretical cut-off frequency in Step 5 , set the oscilloscope trigger level to a voltage within the range of both the input and output signals, and use the equation 𝜙 ௗ௘௚௥௘௘௦ = 360 ∗ 𝑓 ∗ 𝛥𝑡 to calculate the phase lag of the output from your 1 st order system. Note: 𝑓 (Hz) is the frequency you inputted into the function

6 generator and 𝛥𝑡 is the time difference (seconds) you measured with cursors between the input and output peaks (set two cursors in different channels). Record the output phase lag for a set of frequencies until you find an input frequency that gives you a phase lag that is very close to 45° . You will need to include these data in your memo report. This method will not give a super accurate result due to the resolution of the MyDAQ oscilloscope (5 μs). You may need to interpolate your data to determine a more accurate cut-off frequency. Cut-off frequency 𝑓 ௖ (Hz): 9. Close the myDAQ Function Generator and Oscilloscope windows, use the “Bode Analyzer” functionality from the “Instrument Launcher” to plot the circuit’s gain and phase shift as functions of frequency. Change the “Steps” of the Bode plot to a value among 15~20 (marked in red in the figure below ), then click “Run”, and use the cursor (marked in blue in the figure below ) to find and record the cutoff frequency (also known as the corner frequency) of your low-pass filter. This is the frequency corresponding to 45° phase lag, and it is also the frequency at which the Bode magnitude is 3 dB below its maximum value. You may interpolate your data to determine a more accurate cut-off frequency. Bode cut-off frequency 𝑓 ௖ (Hz): Take a screenshot of the Bode plot. You need to include the Bode Plot in your memo report and mark the cutoff frequency on the plot. 10. Show all your time constant and cut-off frequency results to your TA and obtain his/her signature, before taking apart your circuit.

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7 𝑅 ଵ 𝑅 ଶ 𝑅 ଷ 𝑅 ௜ 𝑅 ସ 𝑅 ହ 𝐶 𝑒 ଵ 𝑒 ଵ 𝑒 ଶ 𝑒 ଶ 𝑒 ௢ 𝑒 ௢ 𝑒 ௜ 𝑅 ଵ 𝑅 ଶ 𝑅 ଷ 𝑒 ଵ 𝑒 ௜ 𝑒 ௢ 𝐶 𝑅 ௜ 𝑅 ସ 𝑅 ହ 𝑒 ଶ 𝑒 ଵ = − ቆ𝑒 ௜ ൬ 𝑅 ଷ 𝑅 ଵ ൰ + 𝑒 ଶ ൬ 𝑅 ଷ 𝑅 ଶ ൰ቇ 𝑒 ଶ = − 𝑅 ହ 𝑅 ସ 𝑒 ௢ 𝑒 ௢ = − 1 𝑅 ௜ 𝐶 න 𝑒 ଵ 𝑑𝑡

8 Figure 6 . Example of a first order step response. The time constant 𝜏 is the time for a system's step response to reach 1 − 𝑒 ିଵ ≈ 63.2% of its final (asymptotic) value. (http://en.wikipedia.org/wiki/Time_constant)