ELEC105L Fundamentals of Electrical Engineering Spring 2010ELEC105L Fundamentals of Electrical Engineering Spring 2010Lab #9: RC Circuits and Time ConstantsWhy is this important?Combinations of resistors and capacitors have been used for decades in many important timing applications. The timing of most traffic lights in the past, and many in the present, were and are controlled by RC circuits. The timing of the turn signals in most automobiles is controlled in the same way. A primary property of RC circuits, the time constant, is one of the most important concepts in electrical engineering and in many other fields. In this lab exercise, you will work with an RC circuit and measure itstime constants using an oscilloscope.Lab ReportIn a well-organized, professional-style document, outline the design steps, measurements, predictions, and explanations you are asked to complete during this lab exercise. You should provide enough details and background information so that a technically competent but uninvolved reader could understand the results you are presenting. Remember to include other details that are helpful to readers, such as an introduction and conclusion, diagrams, units on values, annotations on graphs, well-defined variables, etc. Your format should be that of a progress report, so your calculations, results, and procedures should be presented in a chronological, integrated format, not in separate sections. You are strongly urged to consult the “Lab Report Guidelines” available at the lab web site. Each group should submit a single report with the names of all of the students who worked together, the lab number, and the date the report was submitted. Final versions of reports are due at the beginning of the lab session next week; there will be no peer review cycle.Lab reports are weighted 100 points each.BackgroundWhen capacitors are present in circuits that are powered only by one or more DC sources, they tend to be charged to a particular voltage determined by the rest of the circuit and remain charged at that voltage as long as the circuit does not change state. A change in state could occur if someone were to switch the power on or off or if a second switch were to apply or remove power to part of the circuit. For example, when the slide switch in Figure 1 below is in the indicated position, current flows through the two resistors at all times. However, if a long time (say, several seconds) has passed since the switch was placed in that position, there is not likely to be much, if any, charge moving into or out of the capacitor. In that case, we say that the capacitor is charged and that the circuit as a whole is in equilibrium.Figure 1. A simple RC circuit. When the slide switch is in the indicated position, the capacitor charges (i.e., voltage vC rises until an equilibrium state is reached). When the switch is moved to the right-most position, the source of power is disconnected, and the capacitor is free to discharge through resistor R2.A general expression that describes how the voltage across a capacitor changes in response to a sudden change in the circuit it is in is given byslideswitch+−CR1vsR2+vC(t)−−  RCtfifCevvvtv,where vi is the “initial” voltage across the capacitor at the instant the circuit changes state (such as the slideswitch in Figure 1 moving to the right), and vf is “final” voltage across the capacitor after the circuit reaches its new state of equilibrium. The expression above is valid for t > 0, where t is understood to be the amount of time that has passed since the change in state took place, and it applies whether the capacitorcharges or discharges, that is, whether vf > vi or vice versa. The resistance R in the exponent is the Thévenin equivalent resistance “seen” by the capacitor with the circuit in its new state. The product RC is the time constant (in seconds) that dictates how quickly the capacitor charges or discharges.For example, if the switch in Figure 1 were moved to the right at time t = 0, then the expression that describes how the capacitor voltage vC changes with time as a result of that state change would be given by     CRtcCRtcCevevtv220000,where we have recognized that vf = 0 because the capacitor will eventually discharge to zero volts after enough time has passed, and that vi = vC(0) is the voltage across the capacitor at the instant t = 0. Remember that the voltage across a capacitor cannot change instantaneously, so the capacitor voltage just after the switch is moved is the same as the voltage just before the switch is moved. Before t = 0, the circuit is in equilibrium, and the capacitor voltage is given by, using the voltage divider formula,2120RRRvvstC.If the switch were to be kept in the rightward position for a long time (say, at least ten time constants), the voltage across the capacitor would eventually drop to zero. Of course, if the switch were to be moved backto the left, the capacitor would charge again. However, this time the applicable time constant would be different from the one that regulates the discharging case. The charging case will be considered below.Procedure1. Construct the circuit shown in Figure 1 using the bench-top power supply to serve as vs, and set it to 10 V. The capacitor value should be C = 22 F. Be sure to observe the correct polarity of the capacitor. It is an electrolytic type, and because of its construction, one terminal must always be at a more negative voltage than the other terminal. The negative terminal is marked on the capacitor’s body. The values of resistors R1 and R2 should be chosen according to the following two constraints:a. The capacitor should charge to a final value of 8 V after the circuit reaches equilibrium with the switch in the leftward position.b. The capacitor voltage should drop from 8 V to 4 V one second after the switch is moved tothe rightward position.Clearly explain in your report how you arrived at the two resistor values. To do this, you will need to extend the analysis a little in the “Background” section above.2. Using the single trace feature of the oscilloscope, display the capacitor voltage vC vs. time as it goes through a charge cycle, starting in the fully discharged state with vc = 0. Before you begin, set the vertical scale (V/div) and horizontal scale (sec/div) to appropriate values for the results you