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MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Department of Physics 8.02 Fall 2003

Experiment 7: Time Constant of an RC Circuit OBJECTIVE To show that the theoretical curves we derive in class for the charging and discharging of a capacitor actually apply to the real world! (or, how does Nature know the value of e?) To measure the time constants associated with a discharging and charging RC (resistive-capacitive, or resistor-capacitor) circuit. To use auxiliary features of DataStudio for enhanced data analysis of exponential decays. INTRODUCTION A. Capacitor (See the 8.02T Study Guide, Section 5.1, for a more extensive discussion of capacitors and capacitance.) Capacitors are circuit elements that store electric charge Q , and hence energy, according to the expression ,Q C V= (7.1) where V is the voltage across the capacitor and C 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 have capacitances in the range between picofarads ( )121pF 10 F−= and millifarads ( )31mF 10 F 1000 F−= = µ . Note that we’ve also used

the notation for a microfarad, -6 -31µF=10 F =10 mF.

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B. Charging a Capacitor Consider the circuit shown in Figure 7.1. The capacitor is connected to a DC voltage source of emf E . At t = 0, the switch S is closed. The capacitor initially is uncharged with ( )0 0q t = = . (In the following discussion, we’ll use the convention that represents a time-varying charge as “q” instead of “Q”.)

Figure 7.1 (a) RC circuit (b) Circuit diagram for t < 0 (c) Circuit diagram for t > 0 The expressions for the charge on, and hence voltage across a charging capacitor, and the current through the resistor, are derived in the 8.02T Study Guide, Section 7.6.1. These instructions will use the notation RCτ = for the time constant of either a charging or discharging RC circuit.

Figure 7.2 Voltage across capacitor as a function of time

The capacitor voltage as a function of time is given by

( ) ( ) ( )/1 tC

q tV t e

Cτ−= = −E ; (7.2)

a graph of this function is given in Figure 7.2.

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The current that flows in the circuit is equal to the derivative with respect to time of the charge,

0t tdqI e I e

dt Rτ τ− − = = =

E (7.3)

where 0I is the initial current that flows in the circuit when the switch was closed at t = 0. The graph of current vs. time is shown in Figure 7.3:

Figure 7.3 Current as a function of time

After one time constant τ has elapsed, the voltage has increased by a factor (1−e−1) = 0.632 ; ( ) ( )11 0.632CV eτ −= − = EE (7.4) and the current has increased by a factor of 1 0.368e− = , ( ) 0τ 0.362I I= . (7.5) . D. Discharging a Capacitor Suppose we initially charge a capacitor to a charge 0Q through some charging circuit. Suppose at time 0 t = the switch is closed (Figure 7.4). The capacitor will begin to discharge.

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The expressions for the charge on, and hence voltage across a discharging capacitor, and the current through the resistor, are derived in the 8.02T Study Guide, Section 7.6.1.

Figure 7.4 RC circuit with discharging capacitor The voltage across the capacitor in a discharging RC circuit is given by

( ) ( ) 0 tC

q t QV t eC C

τ− = =

. (7.6)

A graph of voltage across the capacitor vs. time for the discharging capacitor is shown in Figure 7.5:

Figure 7.5 Voltage vs. time for discharging capacitor The current also exponentially decays in the circuit as can be seen by differentiating the charge on the capacitor;

( ) 0 tQdqI t edt RC

τ− = − =

. (7.7)

A graph of the current flowing in the circuit as a function of time also has the same form as the voltage graph depicted in Figure 7.6.

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Figure 7.6 Current vs. time for discharging capacitor EXPERIMENT OVERVIEW In this experiment, you will assemble circuits with resistors and a capacitor, and apply a voltage in a manner that alternately charges the capacitor and allows the capacitor to discharge. DataStudio will be used to determine the time constant of the circuits, both graphically and analytically. Three resistor combinations will be used:

1. A single 100-Ω resistor in series with a 330-µF capacitor 2. Two 100-Ω resistors in parallel, with the parallel combination in series with a

330-µF capacitor. 3. Two 100-Ω resistors in series with a 330-µF capacitor.

APPARATUS/DataStudio SET-UP A. 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-Ω resistor (color code brown, black, brown) and the 330-µF capacitor; connect the Current Sensor in series with the other circuit elements forming a closed circuit using the 750 Interface as the voltage source. Connect the Current Sensor directly into the Analog Channel A on the 750 Interface.

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3. In order to measure the voltage across the capacitor, you must connect the Voltage Sensor in parallel with the capacitor. Connect the Voltage Sensor directly into the Analog Channel B on the 750 Interface.

B. Computer: Connect the 750 Interface to your computer, turn on the Interface, and then turn your computer on.

C. Data Studio File: Download the Data Studio file “exp07.ds ” from the web page and save it on your desktop. Open the activity by double clicking on the icon on the desktop. Your file should have a Signal Generator Display, and a single Graph Display which is already set up to display Current vs. Time and Voltage Across Capacitor vs. Time. If the graphs are not present, see part A of DATA ANALYSIS, below. When you take data you should see something like this

V across C

Current

D. Signal Generator:

1. In the Signal Generator dialog (Figure 7.8) we have chosen “Pos(itive) Square Wave Function.”

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2. The Amplitude has been adjusted to 4.000 V , the Frequency to 0.400 Hz and the sampling rate to 1000 Hz . 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.

Figure 7.8 Signal Generator display E. Current and Voltage Sensor Setup:

1. Click the Setup button. On the Sensor menu, drag the Current Sensor icon and place it on the Analog Channel A.

2. Click the Setup button. On the Sensor menu, drag the Voltage Sensor icon and

place it on the Analog Channel B.

Figure 7.9 Current and Voltage Sensor Setup

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DATA RECORDING A. Graphs: 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. B. Sampling Options: Click on the drop-down menu labeled “Experiment” on the top tool bar. In the “Experiment” menu, click on “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 3.5 seconds in the window. If these options are not set in this manner, set them to these values. C. Data Recording: Press “Start” to begin taking data. Once the data has been recorded, scale the plots to fit the graph screens 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 In this experiment, you are asked to measure the time constants for the three RC circuits described above in EXPERIMENT OVERVIEW. In setting up the apparatus, you first should record data for a single 100-Ω resistor in series with the 330-µF capacitor. Answer the questions for this data, and then repeat the measurement of the time constant for the other two circuits, as described on page E7-5. The first time you measure the time-constant, we want you to use both methods of determining the time constant described below. For the next two circuits, you can use the method you like best, but state on your tear-sheet which method you use.

There are several ways to measure the time constant for the RC circuits. We describe two methods below: Method 1: The current in the charging circuit with initial value I0 at t = 0 decreases exponentially in time, ( ) 0 0

t RC t τI t I e I e− −= = , where τ = RC is the time constant, as described above and in the 8.02T Study Guide, Section 7.6. You can determine the time constantτ graphically by measuring the current ( )1I t at a fixed time t1 and then finding the time t1 + τ such that the current has the value ( ) ( ) ( )1

1 1 10.368I t I t e I tτ −+ = = (7.8)

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Figure 7.10 Current as a function of time in a discharging RC circuit. In the Current, ChA graph, enlarge the Graph window by clicking and dragging anywhere on the edge of the graph window. Click on the “Zoom Select” (fourth from the left) icon in the Graph icon bar and form a box around a region where there is exponential decay for the current. Click on “Smart Tool” (sixth from the left) icon. Move the crosshairs to any point (at some time t1) on the exponentially decaying function and record the values of the current and the time. Multiply the current value (displayed in the Smart Tool feature) by e−1 = 0.368. (If you don’t have a familiar calculator with you, the laptop should have this feature; go to “Start” at the lower left, and follow the prompts through “Accessories” and “Calculator”. DataStudio does have a calculator feature, but its use may seem somewhat cumbersome.) Use the Smart Tool to find the new time t1 + τ such that the current is down by a factor of e−1 = 0.368. Determine the time constant and record your value. Questions First Circuit– Method 1 (answer on the tear-sheet at the end!!!):

1. What is your measured value using Method 1 for the time constant for this first circuit (a single 100-Ω resistor in series with a 330-µF capacitor).

2. What is the the theoretical value of the time constant for your circuit? 3. How does your measured value compare to the theoretical value for your circuit?

Method 2: A second approach is to take the natural logarithm of the current, using the facts that ( )ln te tτ τ− = − and ln( ) ln lnab a b= + . This leads to ( )( ) ( ) ( ) ( ) ( )0 0 0ln ln ln ln lnt tI t I e I e I tτ τ τ− −= = + = − . (7.9)

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Thus, the function ( )( )ln I t is a linear function of time. The y-intercept of this graph is ( )0ln I and its slope is slope 1 τ= − . Thus, the time constant can be found from the slope according to 1/ slopeτ = − (7.10) 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 ( )xln . Then, with the “ x ” highlighted, scroll down the Scientific options again to select ( )xabs . 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 then 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 (most of which is no use to us, as the current is so small for most of the run) will appear (see Figure 7.11 below). Use the “Zoom Select” to isolate the small amount of 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. Use your value of the slope to calculate the time constant.

Useful data

Figure 7.11: The ln(I) versus t plot of all the data. The useful data is indicated.

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Questions First Circuit – Method 2 (answer on the tear-sheet at the end!!!):

1. What is your measured value using Method 2 for the time constant for this first circuit (a single 100-Ω resistor in series with a 330-µF capacitor).

2. How does this Method 2 measured value compare to the theoretical value for your

circuit? Questions Second Circuit– Use Either Method 1 or 2 (answer on the tear-sheet at the end!!!):

1. What is your measured value using the method of choice for the time constant for this second circuit (Two 100-Ω resistors in parallel, with the parallel combination in series with a 330-µF capacitor). State your method of choice.

2. What is the theoretical value of the time constant for your circuit? 3. How does your measured value compare to the theoretical value for your circuit?

Questions Third Circuit– Use Either Method 1 or 2 (answer on the tear-sheet at the end!!!):

1. What is your measured value using the method of your choice for the time constant for this third circuit (Two 100-Ω resistors in series with a 330-µF capacitor). State your method of choice.

2. What is the theoretical value of the time constant for your circuit? 3. How does your measured value compare to the theoretical value for your circuit?

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY

Department of Physics 8.02 Fall 2003

Experimental Summary 7: Time Constant of an RC Circuit Group ___________________________________

Names ____________________________________

____________________________________ ____________________________________

Questions First Circuit– Method 1 and 2 (answer on the tear-sheet at the end!!!):

1. Enter your measured values and the predicted value for the time constant of the first circuit (a single 100-Ω resistor in series with a 330-µF capacitor) in the table below. Method 1: _____________ Method 2: _______________

Predicted value: _______________

Method 1 value/Predicted value: ________________

Method 2 value/Predicted value: ________________

2. Which method, 1 or 2, do you think gives the best results? Why?

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Questions Second Circuit– Use Either Method 1 or 2:

1. What is your measured value using your method of choice for the time constant for this second circuit (Two 100-Ω resistors in parallel, with the parallel combination in series with a 330-µF capacitor). State your method of choice. Answer: _______________________ sec

2. What is the theoretical value of the time constant for your circuit?

Answer: _______________________ sec

3. How does your measured value compare to the theoretical value for your circuit?

Measured Value/Predicted Value: __________________________

Questions Third Circuit– Use Either Method 1 or 2:

1. What is your measured value using your method of your choice for the time constant for this third circuit (Two 100-Ω resistors in series with a 330-µF capacitor). State your method of choice. Answer: _______________________ sec

2. What is the theoretical value of the time constant for your circuit?

Answer: _______________________ sec

3. How does your measured value compare to the theoretical value for your circuit? Measured Value/Predicted Value: __________________________