Phys 31220 Fall 2012 RC CIRCUITS Introduction: In this experiment the rates at which capacitors in series with resistors can be charged and discharged are measured directly with a picoscope (scope) and the RC time constants are found. To make the measurements, the output voltage of a function generator, which is set to generate square waves is applied to the RC circuit. The scope is used to measure directly the time variation of the voltage across the capacitor while it is charging and discharging. From these measurements, the time constants of the RC circuits can easily be determined. As well as illustrating the time variation of charge in RC circuits, the experiment provides an introduction to the use of a picoscope. A scope provides a visual display on the screen of the input voltage as a function of time. The measurements themselves are relatively straightforward, but it takes a little time to become familiar with all the different settings of the scope, which displays the voltage as a function of time. You will be given information about the proper initial settings for the controls. Equipment: Square-wave function generator, Picoscope Computer Interface, digital multimeter, 2R-2C circuit board, wires for circuit hookup, computer running Picoscope Software. Physics Introduction: (a) Charging a capacitor When a constant voltage (potential difference) εo is applied at time t 0 to a series RC circuit (a series combination of a resistor and a capacitor) it takes time to charge the capacitor. If the capacitor is initially uncharged, the charge Q on the capacitor grows with time as Q(t) Qo {1 e t/τ } and the current I in the circuit decreases with time as I(t)Ioe t/τ. Here Qo εoc is the final equilibrium charge on the capacitor which is approached at large time t and Io εo/r is the current at t 0. The constant τ is called the (capacitive) time constant for the circuit and has the value τ RC. Since the voltage across a capacitor is always V Q/C, the voltage across the capacitor in the circuit varies with time as V(t) o{1 e t/τ }. ε (b) Discharging a capacitor If the applied voltage is suddenly reduced to zero at a later time (a new t 0) when the charge on the capacitor is Q1, the charge on the capacitor and the voltage across the capacitor decay (decrease) with time as Q(t) Q1 e t/τ and, V(t) V1 e t/τ (Q1/C) e t/τ, and the current (opposite direction from the original current) in the circuit decreases as I(t) I1 e t/τ with I1 Q1/RC. 6-1 1
V a Square Function Generator R I 600 Ω a R E C V C Picoscop V 0 A square-wave function generator will be used to apply and remove a potential difference Vo εo across the RC circuit under study. The function generator switches the voltage repeatedly from V 0 to V Vo εo and back at the frequency ν of the square wave as shown in the top graph. The period το of the wave is 1/ν. Since the voltage across C varies as V(t) V o{1 e t/τ }, it starts at V 0 at t 0 and rises to V Vo{1 e 1 } 0.632Vo at t τ. The time constant τ may be determined directly by measuring how long it takes for the voltage to rise from V 0 to V 0.632Vo. This is done by observing the voltage V as a function of time with the picoscope. The trace on the scope screen represents a plot of V as a function of t. Since you will be interested in the shape of the curve, i.e. its time dependence, rather than the actual magnitude of the voltage (which depends on the circuit parameters and the applied voltage), it will not be necessary to measure V in volts. Instead, you can simply measure V directly on the screen of the picoscope as the vertical displacement (in scale divisions on the scope screen) of the scope trace from the baseline (which corresponds to zero volts, or ground). It is necessary to know the number of vertical scale divisions on the screen corresponding to Vo. To determine this, you must apply the constant voltage to the RC circuit for a time long enough for the charge to approach its maximum value. The time constant τ can also be determined by measuring how long it takes for the voltage to drop from V to V 0.368Vo. This is done by observing the voltage V as a function of time with the picoscope. IMPORTANT: Setting up the Picoscope Connect the square wave function generator to the scope (600 Ω to red and Ground to black of input). Turn on the function generator and make sure it is set for square wave output. Go through the following steps to initialize the picoscope. (If the scope freezes at any time close it and reopen it and follow pt. 2 again) 1. Make sure the Computer is running windows. If it is not, then hold down the option button and restart the machine. Keep the option button depressed till you reach a screen asking you to choose between windows and Mac. 2. On the Windows Desktop you will find the picoscope icon (with the picture of a wave). Open Picoscope, maximize it to fill the screen and check the following: a. The Trigger (bottom right) should be set to Auto. b. Go to Settings Options and choose Current(Filtered) in the Display Mode box c. Set the Time Base (upper right) to 50 µs. 6-2 2
Test Run Observe the square wave on the scope screen. The top graph in the figure at right shows what the square-wave output Va of the function generator should look like. Try changing the frequency (by selecting different ranges and turning the fine tune knob) and amplitude of the square wave and see how the pattern changes. Try to adjust the picoscope time base (top right) so that you see about one complete wave in the picoscope. Square Wave (Function Generator) Frequency is too high!!! Correct Frequency. MEASUREMENTS: A. Record 2R-2C Board data Measure the resistances of the two resistors and capacitors with the multimeter and record them. Remember to turn the dial of the multimeter and set it to measuring resistance or capacitance accordingly. Use the Multimeter Data Sheet (http://www.nd.edu/~ssaddaw1/multimeter%20data%20s heet.doc) to determine the error in measuring these quantities V V a a Square Wave R E Function Generator TIME CONSTANT FOR SMALL R & SMALL C Select the smaller resistance as R E and the smaller capacitance as C. Connect the output of the function generator in series with R E and C and connect the scope across C as shown in the above schematic circuit diagram. Warning: the scope and function generator have internal grounds of their own. If the circuit is to function properly the Ground terminal of the function generator and the grounded scope input (black) lead must be connected to the same point in the circuit (marked V 0 in the diagram). R I 600 ž V 0 Scope 6-3 3 C V V C
Note: that the function generator has an output impedance (effective resistance) R I of about 600Ω. This acts as an additional resistance in the series RC circuit so the total resistance is R T R E + R I and the time constant is τ R T C. Assume that R I 600Ω. Adjust the frequency of the function generator to a value sufficiently low that the voltage Vc approaches the value V o asymptotically as shown for Case (2) in the figure. If you use a frequency which is too high, you will observe a waveform such as that shown for Case (1), and it will be very difficult (if not impossible) to determine the time when Vc 0.632V o. Adjust the horizontal (time axis) scale of the display using the "time base" (TOP RIGHT) controls on the picoscope so that you can see at least one full cycle for V as a function of time(as shown in the figure above). Print the graph on the Picoscope from the File menu on the top toolbar. Measure, if you drag your mouse horizontally in the picoscope graph, a horizontal line will be created which will help you do this. Clicking close to this line will make it move up or down. Position this line at the top of a crest and measure. Alternatively, you can adjust the Amplitude knob in the frequency generator to get a desired voltage like 6V on the graph. B. Time Constant for Charging Locate the point where V is a minimum and is beginning to rise sharply. Create a horizontal line in the picoscope where the voltage is a minimum. Create a vertical line here by dragging the mouse vertically. Record the x(time, t 1 ) and y(voltage) coordinates that appear at the top of the screen. Create a horizontal line at 0.632 of this rising curve. Try to make it as exact as possible by noting the y coordinate readings at the top of the screen and adjusting the position of the line. Create a vertical line where the horizontal line meets the curve. Record the coordinates(t 2, V 2 ). CAUTION: Clicking next to a line moves it so be very careful where you click on the picoscope. The time constant τ is given by (t 2 - t 1 ). C. Time Constant for discharging. Locate the point where V is a maximum and is beginning to fall sharply. Create a horizontal line in the picoscope where the voltage is a maximum. Create a vertical line here by dragging the mouse vertically. Record the x(time, t 1 ) and y(voltage) coordinates that appear at the top of the screen. Create a horizontal line at 0.368 of this falling curve. Try to make it as exact as possible by noting the y coordinate readings at the top of the screen and adjusting the position of the line. Create a vertical line where the horizontal line meets the curve. Record the coordinates(t 2, V 2 ). The time constant τ is given by (t 2 - t 1 ). TIME CONSTANT FOR LARGE R & LARGE C Reconnect the circuit using the larger of the two capacitances and the larger of the two resistances Adjust the frequency of the frequency generator(source) and the time base of the picoscope so that you get a graph similar to the previous one. Print the graph Follow procedures B and C to get the time constant for charging and discharging for this circuit. 6-4 4
Analysis: (Do this for both the circuits) Using your measured value for R E and C, compute the total resistance R T R E + R I. Calculate the capacitive time constant τ R T C for this circuit from your data. Calculate an approximate error for this value of τ assuming a 2% error in C and a 0.5% error in R T. Compare your measured and calculated values of τ. Do they agree reasonably well when you consider your error estimates? Questions for the report: For each circuit do the time constants τ for charging and discharging the capacitor, which you measured using the picoscope, agree as expected? Does your calculated value of τ for this RC circuit agree within errors with your measured values? Checklist: Picoscope Plot from the Small R & Small C Circuit Picoscope Plot from the Large R & Large C Circuit Data Sheet (initialed by your TA) 6-5 5
A. Data on 2R-2C Board DATA TABLE FOR RC CIRCUITS Element Resistance (ohm) δr Element Capacitance(µF) δc R 1 C 1 R 2 C 2 Time Constant for smaller R and smaller C. R T R E + R I τ R T C δτ Charging: 0.632 x 1 (t 1 ) y 1 (V 1 ) x 2 (t 2 ) y 2 (V 2 ) Discharging: 0.368 x 1 (t 1 ) y 1 (V 1 ) x 2 (t 2 ) y 2 (V 2 ) τ (t 2 - t 1 ) τ (t 2 - t 1 ) Time Constant for larger R and larger C. R T R E + R I τ R T C δτ Charging: 0.632 x 1 (t 1 ) y 1 (V 1 ) x 2 (t 2 ) y 2 (V 2 ) Discharging: 0.368 x 1 (t 1 ) y 1 (V 1 ) x 2 (t 2 ) y 2 (V 2 ) τ (t 2 - t 1 ) τ (t 2 - t 1 ) 6-6 6