Lab_3_3130

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z ELEX 3130 – Signal Processing and Power Electronics Lab number: 3 Response of 2 nd Order Circuits in the Time-Domain Philip Krol A01271678 October 13, 2022 Objectives To analyze the behavior of second RLC circuit by observing the time response with underdamped, critically damped, and overdamped circuit parameters. Equipment:

Resistors: 10 Ω, 100 Ω, 1 kΩ for different levels of damping Inductors: 1 mH Capacitors 100 nF Pre Lab calculations:

Procedure Note: for this lab you will use your DC power supply as the source for this test as your function generator has a (relatively) significant series resistance (nominal 50 ) that will Ω affect the damping factor of your system. 1. Implement your series RLC circuit with = 10 , = 1 mH and = 𝑅 Ω 𝐿 𝐶 100 nF on your breadboard, taking the capacitor voltage as the output. 2. Implement a method of disconnecting the source to/from the supply (this will allow you to check your circuit’s step response). You may wish to use a switch or just directly connect the supply to your breadboard. Note: If using a switch, ensure your circuit behaviour is not affected by contact bounce. 3. Set the DC supply to 5 V and set the current limit to 0.5 A to prevent the current limiting feature of the power supply from interfering with your step response. 4. For 𝑅 = 10 , Ω 𝑅 = 100 Ω and 𝑅 = 1 k , Ω apply a 5 V step input by connecting the supply to your circuit using the method determined in Step 2. a. Capture the output voltage (capacitor) versus waveform with the same time base and compare with your predictions. b. Using cursors, measure the rise time of the step response, i.e., the time taken to go from 10% to 90% of the final value

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For Resistor = 10 Ohms For Resistor = 100 Ohms

For Resistor = 1KOhms C. Determined the percentage overshoot using your measurements (if applicable) D. Approximate the settling time using your measurements (if applicable) 5. Calculate the resistance required to achieve a critically damped step response then implement in your circuit (approximate resistance is OK or you may use a potentiometer for exact values). Repeat Step 4 with this new value of resistance in your circuit. R = 200 Ohm

6. Rearrange your circuit, now taking the inductor voltage, 𝑣𝐿 ( 𝑡 ), as the output. For 𝑅 = 10 apply a step input by connecting your source using Ω the method determined in Step 2. Capture or plot this inductor voltage versus time. V_L at R = 10 Ohms

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Tip: To see the expected response, your step input should be applied when the capacitor has fully discharged. Previously your capacitor was able to discharge (slowly) through the internal resistance of your oscilloscope however you are now no longer using the probe to measure across the capacitor itself, so it may no longer be doing so. You should be sure to fully discharge your capacitor before capturing your step response. This tip also applies to Step 7. 7. Devise a method of measuring the series RLC circuit current, ( ), and, 𝑖 𝑡 for = 10 , apply 𝑅 Ω a step input by connecting your source using the method determined in Step 2. Capture or plot the current waveform versus time.

1. Now, with the capacitor voltage at the output again and = 10 , change 𝑅 Ω the source voltage to your function generator now set up with a square wave which ranges from 0 to 5 V with a frequency of 10 Hz. Set up your scope so you can see the step response from this square wave input a. Note down how this case differs from earlier when using the DC power supply as your source (from Step 4 with 𝑅 = 10 ) Ω and suggest reasons for any difference. The capacitor will discharge depending on the initial input voltage due tp the charging of the capacitor. The output will be a square shaped wave b. Increase the square wave frequency up to 5 kHz and adjust the pulse width (duty cycle) of the square wave from 70% to 80%. With this adjustment, the starting voltage of your capacitor at the rising edge of the pulse should be noticeably different, i.e., in the 80% case we begin to charge at the negative peak or point of “undershoot”, creating a different initial

condition for the step response of the circuit. How does this adjustment affect the overshoot observed in your circuit? The Zeta value is decreased because of the lower frequency caused by the smaller frequency c. With the duty cycle back at 50%, continue to increase the frequency in 1 kHz increments up to around 200 kHz and note your observations with respect to the peak-to-peak amplitude of your circuit. What do you notice around the natural frequency, = 𝑓𝑛 1 2 √ ? What happens as the frequency increases beyond that point? 𝐿𝐶 How does the behaviour compare with the first order circuit? Would you consider this a “low-pass” or “high-pass” filter? I would consider it a low pass filter, it has a larger amount of cycles because of the increase in frequency a. With the function generator at 200 kHz, adjust the duty cycle between 10% and 90% and note your observations at the output. Little change because it is only at 10% Analysis and Discussion 1. Would the circuit have no damping at all (a damping factor of 0) if you did not include a resistor? Why or why not? I believe the circuit would still have damping caused by the frequency and delta value 2. Based on your results in Step 4 (for all repetitions of this step), classify each as underdamped, overdamped, or critically damped. The 10 ohm & 100 ohm circuits are underdamped, 1k ohm are over damped 3. Although you did not derive the step-response equation for inductor

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voltage, ( ), in 𝑣𝐿 𝑡 class, was the step response for ( ) 𝑣𝐿 𝑡 as you expected? Why or why not? Once our circuit reached steady state, our inductor became a short circuit when the capacitor became fully charged 4. Why does the current in the series RLC circuit ultimately reach at 0 after the step response has settled even though a voltage is still being applied? It will decrease to 0 eventually when the capacitor is fully charged 5. Explain/justify any differences observed in 8a. The voltage across the conductor will rise and decrease depending on if the input voltage is greater than the capacitors current.