Lab 7 - RC Circuits

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Florida International University *

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MISC

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Electrical Engineering

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Apr 3, 2024

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Name ___________________________ Date___________________ RC CIRCUIT SIMULATION Introduction In this simulation you will study the behavior of the current and voltages of a RC circuit. In the diagram above we display an RC circuit. When we close the switch S 1 leaving the switch S 2 open, the capacitor is charged through the resistor by the constant voltage source. The voltages across the capacitor and resistor change exponentially with time. The voltage across the capacitor and the resistor are given as, V C ( t )= V (1− e − t / τ ) V R ( t )= V e − t / τ , where V is the applied voltage and τ=RC is the time constant. Once the capacitor is charged we can open the switch, S 1 and then close the switch S 2 . The capacitor will now discharge through the resistor, the voltages across the capacitor and resistor in this case are given by, V C ( t )= V e − t / τ V R ( t )=− e − t / τ , where V is the initial voltage across the capacitor. The rate at which the capacitor charges or discharges is characterized by the time constant τ = RC. When charging , RC is the time that it takes for the capacitor voltage to increase from zero voltage to 0.632 times the charging voltage, since at t = τ = RC V C ( t )= V (1− e −1 ) = V (1 - 0.368) = 0.632 V, or 63% of V Similarly, when discharging , RC is the time for the voltage to drop to 0.368 times its initial value, since at t = RC V C = V e -1 = 0.368 V, or 36.8% of V

1 – Setup Open the link (https://phet.colorado.edu/en/simulation/legacy/circuit-construction-kit-ac-virtual-lab) and run the java applet (you could try this link instead: http://phet.colorado.edu/sims/html/circuit- construction-kit-ac/latest/circuit-construction-kit-ac_en.html Opening this software you will see a blank canvas On the right hand side you will see a list of elements you can add to your circuit. Click the schematic button on the right hand side of the screen. To create a circuit simply click and drag a circuit element onto your canvas. Circuit elements like batteries, capacitors and resistors when dragged onto the canvas have a starting value. To change this value right click on the element. Under the tools section on the right hand side is a stopwatch and a voltmeter. Click on both of these. The voltmeter has probes which can be dragged to different locations in the circuit to measure the voltage across circuit elements. In this lab we will use it to measure the voltage across the capacitor. The switches in the circuit simulator have a loose wire which can be dragged to close the circuit. Create in the circuit simulator the RC circuit shown on the first page. It does not need to looks exactly like it, just match the connections as shown in the circuit in the figure above. Choose the following values for the resistor, the capacitor and the battery: R = 100Ω, C = 0.2F , V = 5V. 2 – Charging a Capacitor To begin make sure your capacitor is discharged. That is the potential difference or voltage across the capacitor is zero. You can use the voltmeter to check that it is zero. If it is not zero we can discharge the capacitor by closing switch 2 and opening switch 1. Once the voltage is zero leave switch 1 open and open switch 2. Since this is a simulation it also gives you the option to right click on the capacitor and choose discharge or charge capacitor. If you choose to do this for your setup make sure both switches are open so the capacitor will remain in the charged/discharged state until you are ready.

You are going to time the charging of the capacitor, i.e. the time and it takes for the capacitor to reach a specific value Vc . This can be done with your own stopwatch or the stopwatch in the simulation (it is much easier to use your smartphone as the stop watch). When your ready, start you first measurement Vc = 0.5V and repeat (or it might be more convenient to use the lap function on your stopwatch). Make a copy of the the table below to collect your data (no need to turn it in). The voltage increases faster at the beginning. Vc 0.5V 1.0V 1.5V 2.0V 2.5V 3.0V 3.5V 4.0V 4.5V 5.0V Time Plot Vc as a function of time. 1. Looking at the plot, at about what time did the voltage reach 63% of its total voltage? 2. Assume you do not know the value of the capacitor C , calculate C from the knowledge of R and time constant found in question 1. 3. What is the percent error between the calculated values and the exact value of C ? | Approximate Value − Exact Va | Exact Value | Plot in a semi-log format. If you solve V C ( t )= V (1 − e − t / τ ) for e − t / τ and take the natural log of each side, then you can plot y =ln(1− V C / V ) vs and x=t . Draw a single line of best fit through your data. 4. Write your result as y =m x, what is the value of m ? 5. Using your plot and the result from question 4, what value of τ do you find? 6. Again assuming we do not know the capacitance solve for C using your fitted value of τ and the known value of R. 7. What is the percent difference between the calculated value and the known value of C? 3 – Discharging a Capacitor Repeat the set of measurements taken above but now you discharge the capacitor rather then charge it. The capacitor needs to be fully charged before you start. If you have not changed anything your capacitor should be charged from the previous step at 5V. If not you can open both switches and right click on the capacitor and select the charge option. When you are ready, leaving switch 1 open, close switch 2 and begin timing the discharge of the capacitor as it discharges from 5V to 0V. Make a copy of the the table below to collect your data (no need to turn it in). The voltage increases faster at the beginning. Vc 4.5V 4.0V 3.5V 3.0V 2.5V 2.0V 1.5V 1.0V 0.5V 0.0V Time

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Plot Vc as a function of time. 8. Looking at the plot at about what time did the voltage reach 36.8% of its total voltage? 9. Assume you do not know the value of the resistor R, calculate R from the knowledge of C and time constant found in question 1. 10.What is the percent error between the calculated values and the exact value of R ? Plot in a semi-log format. If you solve V C ( t )= V e − t / τ for e − t / τ and take the natural log of each side, then you can plot y =ln( V C / V ) vs and x=t . Draw a single line of best fit through your data. 11.Write your result as y =m x, what is the value of m ? 12.Using your plot and the result from question 4, what value of τ do you find? 13.Again assuming we do not know resistance solve for R using your fitted value of τ and the known value of C . 14.What is the percent error between the calculated value and the exacts value of R ? 4 – Non-Ohmic Materials (this is a continuation about the topics discussed in the Resistor Simulation) Ohm’s Law states a particular relationship between the voltage applied actors a material and the amount of current which passes though it. I = R 1 V This relation is linear in the sense that if you double the voltage also the current will double as the resistance stays constant . T his is not true for all the materials. For example some material have a resistance which increases with the temperature (Joule Heating). 15.As you turn on a light switch, does the resistance of a light bulbs stay constant ? 16.Do you expect a light bulb to follow the Ohm law? Suppose we have a light bulb connected to a battery. We vary the voltage and measure the current passing through the bulb. Our measured data is recorded in the table below with the voltage in Volts and the current in mA. V 1 2 3 4 5 6 7 8 9 10 I 2.61 5.07 7.46 10.8 12.6 14.3 15.6 16.2 16.7 16.9 Make a plot of I vs V (Use Excel or Libreoffice Calc or whatever you prefer) 17.It the relation between current and voltage in the table above linear? 18.Is the relation linear when the temperature of the bulb is still cold?