Physics 8.02T 1 Fall 2001 Measuring the time constant for an RC-Circuit Introduction: Capacitors Capacitors are circuit elements that store electric charge Q according to Q = CV where V is the voltage across the capacitor andc is the constant of proportionality called the capacitance. The unit of capacitance is the farad, [F] = [C]/[V]; 1 farad = (1 coulomb)/(1 volt). Capacitors come in many shapes and sizes but the basic idea is that a capacitor consists of two conductors separated by a spacing, which may be filled with an insulating material (dielectric). One conductor has charge +Q and the other conductor has charge Q. The conductor with positive charge is at a higher voltage then the conductor with negative charge. Most capacitors are in the picofarad [pf], 10 12 F, to the millifarad range [mf], 10 3 F =1000µF. Capacitors can do many things in both ac circuits and dc circuits. Capacitors store energy Capacitors when coupled with resistors can delay voltage changes Capacitors can be used to filter unwanted frequency signals Capacitors are needed to make resonant circuits Capacitors and resistors can be combined to make frequency dependent and independent voltage dividers In these experiments you will measure the time constants associated with a discharging and charging capacitor in series with different resistors.
Physics 8.02T 2 Fall 2001 Computer Setup: Connect the 750 Interface to your computer, turn on the Interface, and then turn your computer on. Data Studio File: Download the Data Studio file RC Circuits.ds from the Current Assignment webpage and save it on your desktop. If there is already a file by this name on your desktop, save over it, as it may not be set up properly. Open it by double clicking on it. Your file has a Signal Generator Display, and two Graph Displays which are already set up to display Current versus Time and Voltage versus Time. Signal Generator: 1. In the Signal Generator dialog we have chosen Postive Square Wave Function. 2. The Amplitude has been adjusted to 4.0 V, the Frequency to 0.4 Hz, and the sampling rate to 1000 Hz 3. We chose the output data that you will record by clicking the plus button (+) beside Measurements and Sample Rate on the Signal Generator dialog and clicking the appropriate Measure Output Voltage and Measure Output Current buttons. Voltage Sensor Setup 1. Connect the Voltage Sensor directly into the Analog Channel B on the 750 Interface.
Physics 8.02T 3 Fall 2001 2. Click the Setup button. On the Sensor menu, drag the Voltage Sensor icon and place it on the Analog Channel B. Current Sensor Setup 1. Connect the Current Sensor directly into the Analog Channel A on the 750 Interface. 2. Click the Setup button. On the Sensor menu, drag the Current Sensor icon and place it on the Analog Channel A. Graph: Here s how to set up the two graphs if you ever need to (it should already be set up for you here). Drag the Voltage, ChB icon in the Data Window and drag it into the Graph icon. This will create a Voltage (ChB) vs. Time graph. Grab the Current, ChA icon in the Data Window and drag it into the Graph icon. This will create a Current (ChA) vs. Time graph. AC/DC Electronics Lab circuit board 1. Connect the banana plug patch cords from the OUTPUT ports of the 750 Interface to the banana jacks on the AC/DC Electronics Lab circuit board. 2. In order to measure the current that flows in the circuit, you must connect the Current Sensor in series with the 100 ohm resistor (brown, black, brown) and the 330 microfarad capacitor. Put alligator clips on the Current Sensor banana jacks and connect it in series with the other circuit elements forming a closed circuit using the 750 Interface as the voltage source. 3. In order to measure the voltage across the capacitor, you must connect the Voltage Sensor in parallel with the capacitor. Put alligator clips on the Voltage Sensor banana plugs and connect the sensor in parallel with the capacitor.
Physics 8.02T 4 Fall 2001 Sampling Options: Click on drag down menu labeled Experiment on the top tool bar. In the Experiment menu, click on the Set Sampling Option to open the Sampling Options dialog. Check that the Delay Choice is on None. Check that the Automatic Stop choice is Time with 5.0 seconds in the window. If these are not set in this manner, set them to these choices. Data Recording: Press Start to begin taking data. Once the data has been recorded, scale it to fit the graph screen by clicking on the first icon on the left at the top of the Graph Window, which is the scale to fit icon Data Analysis: Measuring the time constant. There are several ways to measure the time constant for the RC circuit. Method 1: The current in the charging circuit with initial value I 0 at t = 0 decreases exponentially in time, I(t) = I 0 e t R C. This function is often written as I(t) = I 0 e t τ where τ = RC is called the time constant. You can determine the time constantτ graphically by measuring the current I(t 1 ) at a fixed time and then finding the time t 1 + τ such that the current has the value t 1 I(t 1 + τ ) = I(t 1 )e 1 = 0.368I(t 1 ).
Physics 8.02T 5 Fall 2001 Drag the Current, ChA icon from the Data Window and drop it on the Graph icon in the Display window. This creates a new current vs. time graph. Enlarge the Graph window. Click on the Zoom Select (fourth from the left) icon in the Graph icon bar. Form a box around a region where there is exponential decay for the current. Click on Smart Tool (sixth from the left). Record the value of the current and the time for any point (at some time t ) on the exponential decaying function. Multiply the current value by e 1 1 = 0.368. Use the Smart Tool to find the new time t 1 + τ such that the current is down by a factor of e 1 = 0.368. Determine the time constant and record your value Questions 1. Estimate the accuracy with which you can determine this point. 2. What is the error in your calculation for the time constant? 3. How does your measured value compare to the theoretical value for your circuit? 4. Explain why there may be a discrepancy between your experimental result and the theoretical value? Method 2: A second approach is to take the natural log of the current, using the two facts that ln(e t τ ) = t τ and ln(ab) = ln a + lnb, which yield ln(i(t)) = ln(i 0 e t τ ) = ln(i 0 ) + ln(e t τ ) = ln(i 0 ) t τ. Thus, the function ln(i(t)) is a linear function of time. The y-intercept of this graph is ln(i 0 ) and its slope = 1 τ. Thus the time constant can be found from the slope according to τ = 1 slope. Click the Calculator button on the bar menu. In the Calculator window click New. The variable x should be highlighted in the Definition window. Click on the Scientific button and scroll down and click on ln(x). Scroll down on the Variables menu and click on Data Measurement. In the Please Choose a Data Source Window, scroll and click on Current, ChA and click OK. Then click the Accept button in the calculator window. A calculator icon with your equation should appear in the Data window. Drag that calculator icon to the Graph icon in the Display window. A fairly complicated graph will appear due to statistical fluctuations in the data. Use the Zoom Select to isolate data where the function is linear. You should see fluctuations in the data due to approximations associated with the sampling rate. You can use Zoom Select to choose the region where there are the smallest fluctuations. Use the mouse to highlight a region of data. Once you isolated this region, click on the Fit button, scroll down and click to linear fit. Record the value of the slope and the y-intercept. Use your value of the slope to calculate the time constant.
Physics 8.02T 6 Fall 2001 Questions: 1.What can you say about the reliability of this method? 2. How did the slope change as you changed the region of selected data? Different Time Constants: 1. Measue the time constant for two 100 ohm resistors in parallel. 2. Measure the time constant for two 100 ohm resistors in series. Now go to WebAssign and enter your three experimental values for the time constants in the space provided.