# Physics 2306 Experiment 7: Time-dependent Circuits, Part 1

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1 Name ID number Date Lab CRN Lab partner Lab instructor Objectives Physics 2306 Experiment 7: Time-dependent Circuits, Part 1 To study the time dependent behavior of the voltage and current in circuits containing a capacitor, a resistor, and a voltage source ( RC circuits ) To gain familiarity with the operation of two very useful test instruments the digital oscilloscope and the function generator Required background reading Young and Freedman, section 26.4 Introduction In the last two labs, you studied circuits where the current and voltage were constants (independent of time). In this lab, you will study circuits where the current and voltage vary with time. For circuits where the voltage varies in a rapid way, there is an essential piece of electronic instrumentation that is needed. It is called an oscilloscope. Generally it is used to display voltage (on the vertical axis) versus time (on the horizontal axis). The time dependent circuits you will look at in this lab will contain capacitors, resistors and a voltage source. Capacitors are devices that store electric charge (and therefore energy). Consider the series circuit depicted in Figure 1. It contains a resistor and a capacitor. When the switch is in position 1, current flows from the power supply to the circuit, thus charging the capacitor. When the switch is in position 2, the power supply is removed from the circuit, and the capacitor is allowed to discharge through the resistor. 1

2 1 2 + DC power supply _ + VP Figure 1 A simple RC circuit. Initially, consider the switch to be in position 2 and the capacitor in Figure 1 to be uncharged. If the switch is flipped to position 1 to a DC power supply with voltage V 0, the charge will build up according to the charge build-up relationship derived in Section 26.4 of Young and Freedman (equation 26.12). Since the voltage across the capacitor is proportional to the charge on it (V=Q/C, where C is the capacitance), we have: t ( ) 1 0 V t = V e RC (build - up) As discussed in your textbook, there is a characteristic time that it takes to charge (or discharge) a capacitor. This time depends only on the characteristics of the elements of the circuit (the resistor and capacitor) and not on the applied voltage. A convenient place to analyze the time to charge the capacitor is to look at the time when t = RC. When t = RC the voltage equation above becomes, V ( t ) RC = V 0 1 e RC = V 0 1 e 1 ( ) We define this interval of time as the RC time constant of the exponential charging function and give it a special symbol τ, where τ = RC. If R is given in ohms and C is given in farads then τ will be in seconds. A simple evaluation of the equations above shows that after one RC time constant has elapsed, the voltage across the capacitor has built up to 1 e 1 = 0. 63, or 63% of its final value of V 0. As you know from your study of exponential functions the voltage approaches the final value of V 0 asymptotically. Theoretically, after an infinite time the voltage is equal to V 0. From a practical point of view the voltage reaches V 0 after a few time constants have elapsed. This is clear if you look at Table 1 and Figure 2. These indicate the percent of the maximum voltage, V 0, 2

3 that is achieved after a certain number of time constants have elapsed. Note that the curve shown in Figure 2 is a universal curve. Once you know the RC time constant of a circuit, this curve gives you everything you need to know about its time behavior during charge build-up. Exponential Growth % of maximum value 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Time constants Figure 2: Voltage exponential build-up (growth) curve in an RC circuit. Table 1: Behavior of growth and decay curves as a function of the number of time constants. Number of time constants % Decay % Growth % 0.00% % 39.35% % 63.21% % 77.69% % 86.47% % 91.79% % 95.02% % 96.98% % 98.17% % 98.89% % 99.33% % 99.59% % 99.75% % 99.85% % 99.91% % 99.94% % 99.97% 3

4 After the voltage across the capacitor has settled at its asymptotic value, the switch is flipped to position 2. This creates a circuit with just a capacitor and resistor, and the capacitor then discharges through the resistor. The charge on the capacitor will decay according to the charge decay relation derived in Section 26.4 of Young and Freedman (equation 26.16). As above, we will monitor the charge on the capacitor, by measuring the voltage across it. The voltage as a function of time during the decay is given by: V ( t) = V0 e t RC (decay) As with the voltage build-up, the voltage decays to zero with the same characteristic RC time constant. After one RC time constant has elapsed, the voltage across the capacitor has decayed to e 1 =. 37, or 37% of its initial value of V 0. Once again, theoretically it would take an infinite time for the voltage to reach zero, but for practical purposes the voltage reaches zero after a few RC time constants have elapsed. This is clear if you look at Figure 3 and Table 1. Once you know the RC time constant of a circuit, this curve gives you everything you need to know about its time behavior during charge decay. Exponential Decay % of maximum value 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Time constants Figure 3: Voltage exponential decay curve in an RC circuit. 4

5 Name ID number Date Lab CRN Lab partner Lab instructor Physics 2306 Experiment 7: Time-dependent circuits, Part 1 Ph 2306 Experiment 7: Prelab assignment (complete and turn in at the beginning of your lab session) 1. Consider the build-up of the charge on a capacitor as described in the introduction to this lab. Assume that you start with an initially uncharged capacitor, as in Figure 1 and then you close the switch to a battery of voltage V 0. After that time, how many time constants must elapse before the charge reaches 39%, 63%, and 86% of its final value? (You do not need to do an exact calculation here; just estimate from the table or curve in the introduction) 2. For the same case you were considering in problem 1, assume that the capacitor is now fully charged. Now the switch is flipped (in Figure 1) to position 2, and the charge on the capacitor begins to decay away. After that time, how many time constants must elapse before the charge reaches 61%, 37%, and 13% of its initial value (the value right before you flipped the switch to position 2)? (You do not need to do an exact calculation here; just estimate from the table or curve in the introduction). 5

6 3. In lab, you will be building a circuit like that shown in Figure 1 in the introduction, but you will want to flip between position R 1 and position 2 much faster than you can do by hand. Instead, V(t) C you will be using a function generator to deliver an input voltage known as a square wave V 0 GND, which alternately steps between zero and some constant value V 0. This is equivalent to regularly switching between a position 1 (V 0 ) and position 2 (zero) of Figure 1 on page 2 of this writeup. Use the formulae on pages 2-4 of this writeup to predict the time dependent behavior of the voltage across the capacitor (V out in the figure above) assuming the square wave input (V(t) in the figure above) is as shown on the figure above. Sketch your prediction in the lower figure (be sure to label the vertical scale with an appropriate numbers of volts/division). Use R = 560 Ω and C = 1 μf; show your calculations and be sure to indicate clearly what the time constant is for this circuit. As you draw your curve, use what you learned about build-up and decay curves in answering questions 1 and 2 to help you carefully sketch the time dependent behavior. V out Horizontal 1ms/div Vertical 500 mv/div Horizontal 1ms/div Vertical 6

7 Equipment You will use the following equipment: Pasco EM-8656 AC/DC Electronics Laboratory Several wires of varying lengths (5, 10, and 25 cm) (in Ziploc bag) One each of the following resistors: 560 Ω, 100 kω (in Ziploc bag) Capacitors: one 1 μf, two 10 μf, one 100 μf (in Ziploc bag) Three red and three black banana plug banana plug cables Four red and four black banana plug to alligator clip adapters Knife switch (single pole, double throw) BK Precision 1670A Adjustable Power Supply Manual range digital multimeter (DMM) One BNC tee (probably attached to the function generator) One BNC BNC cable Two BNC banana plug cables BK Precision 4017A Function Generator Tektronix TDS 1002 Oscilloscope Since this lab involves many components, you should take a minute to see if everything in the above list is present at your station. If you are missing something, consult with your TA. Activity 1: Learning the basics of the function generator and the oscilloscope In this part of the lab, you will go through some steps to familiarize yourself with the operation of the oscilloscope. You will be using the Tektronix TDS 1002 oscilloscope; it is a two channel digital oscilloscope. The oscilloscope is interfaced to the computer; you will use this feature to print the display of the oscilloscope. You will be using the BK PRECISION 4017A function generator to produce periodic signals. Pictures of each device are shown below with labels to help you find the various knobs as we mention them (see Figures 4 and 5). 7

8 Figure 4: Tektronix TDS 1002 two channel digital oscilloscope. Figure 3: Tektronix TDS 1002 twfig Figure 5: BK Precision 4017A function generator. 1. Turn on the oscilloscope and the function generator. Connect the BNC- BNC cable to the CH1 post on the oscilloscope and to one side of the BNC tee that should be attached to the function generator output. 2. On the oscilloscope, press the DEFAULT SETUP button after the oscilloscope has gone through its initial screens and settled on the final screen with the grid pattern. 8

9 3. Set the function generator to the 100 Hz range using the appropriate frequency range select button. Adjust the COARSE and then the FINE, FREQUENCY knobs to set the output frequency to 100 Hz. The frequency is displayed on the red LED display on the front of the function generator. Note: there is a delay between when you adjust the knob and when the frequency display updates. Turn the knobs more slowly if this delay causes you to overshoot your target frequency. 4. Select the triangle signal option among the function selection buttons on the function generator. Adjust the OUTPUT LEVEL knob to about mid scale. All the other push button settings should be in the OFF position (including the -20 db button; it should be in the OFF (out) position). 5. To obtain a quick look at the signal that the function generator is producing press the AUTO SET button on the oscilloscope. After a few seconds the oscilloscope will display a plot of the voltage as a function of time. This plot is often called a TRACE since the pattern is repeatedly redrawn each time the plot reaches the end of the screen. 6. One of the limitations of the AUTO SET function is that the oscilloscope has no way of telling what type of cable you used to connect the function generator to it. The default setting of the oscilloscope is to assume that you are using a special 10X attenuating probe. But actually, you are using just a simple cable, so it is important tell the oscilloscope not assume a special probe is being used. You do this by pressing the CH1 MENU button and toggling to the 1X setting in the Probe option (found on the right hand side of the digital display). 7. Adjust the function generator OUTPUT LEVEL knob so that the signal displayed on the oscilloscope has a peak to peak amplitude of 10 V. The value of the voltage will be displayed on the screen after the AUTO SET button is pressed (under CH1 PK-PK in the lower left hand corner). (If you don t see it, press the AUTO SET button again.) Question 1-1: How many vertical squares does the signal take up on the display? Since you know that the peak to peak amplitude is 10 volts, how many volts are represented by each major vertical division? Suggestion: you can use the POSITION knob on the VERTICAL control panel on the oscilloscope to move the TRACE up and down for easier measuring of the number of vertical squares. Question 1-2: How many horizontal squares does one period of the signal take up on the display? Since you know the period of this 100 Hz waveform, how many seconds are represented by each major horizontal division? Hint: you can use the POSITION knob on the HORIZONTAL control panel on the oscilloscope to move the TRACE side to side for easier measuring of the number of horizontal squares. 9

11 at the end of the lab to submit with your lab report. Open a copy of the report template.doc document from the class notes folder and save it with a new name to the desktop of your computer. Question 1-7: Follow the instructions below to make a copy of the waveform (triangle wave at 1550 Hz with only a few periods displayed) that you generated for Question 1-6 above. Load the OpenChoice software by clicking on the shortcut and then click the Select Instrument button. A menu will pop up. Choose GPIB0::01:INSTR. (If that choice is not present, then look at the document Oscilloscope_bug_instructions.pdf in the ClassNotes directory for this lab and follow the directions there.) Click on the Screen Capture button and then click Get Screen. After the image is displayed on the computer, click Copy to Clipboard to copy the image to the clipboard and then paste it into the appropriate place in your Word template document. Activity 2: Time Dependent Behavior of RC Circuits with Large Time Constants In this activity, you will observe the charge-up and decay of a capacitor in a simple RC circuit with a large time constant. The circuit you will investigate is shown in Figure DC power supply _ + red lead CH2 black lead Figure 6: Simple RC circuit with a single-pole, double-throw switch that allows a DC power supply to be hooked up at position 1 (for charging the capacitor) or a short circuit hooked up at position 2 (for discharging the capacitor). The voltage across the capacitor is monitored with channel 2 of the oscilloscope. NOTE the polarity of the electrolytic capacitor; be sure that yours is hooked up as shown. (also see Figure 7 to understand the polarity labeling on the electrolytic capacitor). 11

13 clockwise). Turn on the DC power supply and bring the voltage up to 10 volts (the current should read 0.00 A). Prediction 2-2: To set the oscilloscope to read the voltage across the capacitor as it charges you will need to choose a vertical and horizontal scale that will fit the data you expect. Based on the value of voltage you set on the power supply what is a reasonable value of the vertical scale factor (VOLTS/DIV) so that the observed waveform will fill most of the screen? Prediction 2-3: Take a look at Figure 2 in the introduction. What is a reasonable number of time constants to let elapse to insure that all of the interesting behavior of the build-up is observed? Based on your calculation of the time constant in 2-1, how much total time does this number of time constants correspond to? What value for the horizontal scale (SEC/DIV) on the oscilloscope will be needed to display this total time? To view Channel 2 only, press the CH1 MENU button until the 1 on the left hand side of the display goes away. Then press the CH2 MENU button until the 2 on the left hand side of the display is present. Look at the CH2 menu items and make sure they are set as follows; if any of them are set incorrectly, then toggle through the options using the button by the item of interest. Coupling DC BW Limit Off Volts/Div - Coarse Probe 1x Invert - Off Set the VOLTS/DIV knob to the value you got in 2-2 and the SEC/DIV knob to the value you got in 2-3. For frequencies < 10 Hz the oscilloscope is automatically set to SCAN mode and the word SCAN will appear in the top center of the display. The RUN/STOP button will start and stop the recording of data; when data-taking is in progress the word SCAN appears at the top of the screen; when data-taking is stopped the word STOP appears at the top of the screen. Press the RUN/STOP button on the oscilloscope to start recording data and set the switch to position 1. You should see the voltage across the capacitor start to increase from 0. Press the RUN/STOP button again to stop the recording of data when the voltage has clearly leveled off. You will notice that the value of the maximum voltage displayed on 13

14 the oscilloscope is less than the 10 V you set the power supply to. (After you have done it once, you may want to adjust things like the vertical position to get a better display. IF you don t like the trace you got, just hit RUN/STOP and start again). Question 2-4: You will now use the voltage build-up curve to determine the RC time constant. To make more accurate measurements, the oscilloscope has several useful features. The CURSOR button places two lines on the display that can be used to make accurate measurements. Press the CURSOR button. Press the menu button next to the word Source to select CH2. Then press the menu button next to the word Type to toggle through the different cursor types. One press gives two horizontal lines that can be adjusted up and down with the VERTICAL position knobs that allow the measurement of the voltage values. What is the final value that the voltage leveled off at (it will not be as large as the power supply voltage) assuming it started at 0 volts? Question 2-5: Calculate the value of the voltage that corresponds to 63% of the maximum value. Press the menu button next to the word Type to toggle the cursor to a time cursor and measure how long it took to reach this 63% of maximum value. This time elapsed between the two time cursors is the RC time constant; write that value below. Capture an image of the oscilloscope screen showing the waveform and cursors using the OpenChoice software and paste it into your Word template file. Part 2: The capacitor charge decay curve Now you will observe the decay in the voltage across the capacitor as it discharges through the resistor. The switch should still be at position 1, so that the capacitor is still fully charged. Press the RUN/STOP button on the oscilloscope to start recording data and switch the switch position from position 1 to position 2 to let the voltage across the capacitor decay away. Press the RUN/STOP button again to stop the recording of data when the voltage has clearly leveled off at its final value. Question 2-6: You will now use the voltage decay curve to determine the RC time constant. Use the cursor function as you did previously to mark the point on your graph where the voltage has decayed to 37% of maximum value. The time elapsed between the two time cursors is the RC time constant. Capture an image of the oscilloscope screen showing the cursors using the OpenChoice software and paste it into your Word template file. Write the value of your measured time constant here. 14

16 CH1 red lead black lead Function generator R red lead CH2 black lead Figure 8: Simple RC circuit with a function generator operating in square wave mode. It alternates between a voltage to charge the capacitor and zero voltage, so the capacitor can discharge. Figure 9: Picture of digital multimeter showing where to insert the capacitor leads to measure the capacitance. Question 3-1: Based on the R and C values you measured on page 15, what do you expect the RC time constant to be for the circuit in Figure 8? 16

17 Construct the circuit shown in Figure 8. Use the BNC tee to connect the output of the function generator to your circuit AND to CH1 of the oscilloscope (you use a BNC-BNC cable from function generator to CH1 on the oscilloscope and a BNC-banana plug cable from the function generator to the female banana plugs inputs on the circuit board). Use a BNC to alligator clip connector to connect so that you can measure the voltage across the capacitor; plug this in to CH2 of the oscilloscope. Also, note in Figure 8 where the red and black leads from the BNC to banana plug cables should be located. After your circuit is set up, turn on the function generator in square wave mode, and adjust the function generator and the oscilloscope so that display looks like the one shown in Figure 10. To help you set up the function generator and oscilloscope correctly, answer Questions 3-2 through 3-5 below. Figure 10: This figure shows how the channel 1 waveform (the input signal from the function generator) should look when you have the function generator properly adjusted. Question 3-2: What is the period of the square wave represented in figure 10? What frequency should the function generator s be output be set to, to display this frequency? Set your function generator to this frequency. Question 3-3: What is the peak-peak amplitude of the square wave represented in figure 10? Set your function generator s OUTPUT LEVEL knob to produce this amplitude signal. Hint: you may need to push in the -20 db button to get the amplitude shown in the figure. 17

19 Post Lab questions (if there are only 10 minutes left in the lab, then skip to these questions and complete them) 1. Below is the voltage build-up curve for an RC circuit. a. What is the RC time constant of this circuit? b. If the resistor in the circuit has a value of 10kΩ, what is the value of the capacitance of the capacitor? 19

20 2. Below is a square waveform produced by a function generator and captured by an oscilloscope. a. What is the period of the waveform? b. What is the peak-peak amplitude of the waveform? c. What is the frequency that the function generator was set to? 20

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