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Experiment 10 RC and RL circuits: Measuring the time constant. Object: The object of this lab is to measure the time constant of an RC circuit and a LR circuit. In addition, one can observe the characteristics of these two circuits and compare them. Pre Lab Assignment: Refer to the theory below.

1. If the time to achieve ½ the maximum voltage (refer to step 11 of the procedure of the RC circuit) across the capacitor in an RC circuit is 0.045 s, what is the capacitance in the circuit if the resistance is 150 Ω?

2. For an RL circuit, the voltage across the inductor drops from 4.0 V to a constant 1.5 V as a current is established. In addition the voltage across a 4.7 Ω resistor builds to a steady 2.5 V. What is the resistance of this less than perfect inductor? (Refer to step 13 of the procedure of the RL circuit for help with this problem.)

3. If it takes 1.2 ms for the voltage across the inductor to drop from 4.0 V to 2.75 V, what is the time constant for the RL circuit in question 2? Hint: What is the fraction of the total change in voltage across the inductor represented by a drop from 4.0 V to 2.75 V?

4. What is the inductance of the RL circuit in questions 2 and 3? Theory:

RC Circuit

The RC circuit was discussed earlier in the semester. To review: A voltage source will charge a capacitor by moving charge through a resistor until the capacitor is fully charged. The potential difference across the resistor is proportional to the current through the resistor while the potential difference across the capacitor is proportional to the charge on the capacitor. The equations for current through the resistor, and charge on the charging capacitor are

0tVI e

Rτ−

= and 0 1t

Q CV e τ−

= −

where t is time, I is the current in the circuit, Q is the charge on the capacitor, V0 is the source voltage, R is the resistance, C is the capacitance and τ is the time constant and equal to RC. If the voltage source is now disconnected, and the capacitor discharged through the resistor, the potential difference and current through the resistor is now reversed. The current in the resistor and the charge on the capacitor decrease according to

maxtVI e

Rτ−

= and max

t

Q CV e τ−

=

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LR Circuit

The LR circuit, shown schematically in Figure 1 behaves similarly. If we close the switch so the voltage source provides current to the circuit (position 1), Kirchhoff’s loop rule yields

0s LV IR− − =E

where V0 is the source voltage, I is the current in the

circuit, R is the resistance and EL is the induced

emf of the inductor. Substituting for the emf of the inductor gives us

0 0dIV IR Ldt

− − =

We can solve this differential equation readily if we divide by R and then let a new variable x be defined

as 0V IR− . Since 0V

R is a constant,

dx dIdt dt

= − and the

differential equation becomes

0L dxxR dt

+ =

L

Remf1

2

Figure 1. An RL circuit. When the switch is closed toward position 1, a current will begin to flow through the resistor and inductor. If, after a current is established in the circuit, the switch is quickly thrown to position 2, the magnetic field in the inductor will continue to push a current through the circuit until the energy in that field is exhausted.

Separating variables and integrating with x = xo when t = 0, yields

lno

x R tx L= −

First we define LR

τ = , then we rewrite the above equation as an exponential equation

to give us t

o

x ex

τ−

=

We note that I = 0 when t = 0, so 0oVxR

= . We then substitute the values for x and xo

from above and rearrange the equation to give us the final result that

0 1tVI e

Rτ−

= −

where LR

τ = = the time constant of the circuit.

The emf across the inductor is determined by LdILdt

= −E

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0 01 t t

LV V RL e L eR R L

τ τ

τ

− − = − − = − −

E

0

t

L V e τ−

=E

If the switch in Figure 1 is quickly changed to remove the voltage source and connect the inductor directly to the resistor (Position 2), the magnetic field that was established in the inductor will collapse, driving a current through the circuit. The loop equation for the voltages now becomes

0LIR− =E or 0dIIR Ldt

+ =

We separate variables and integrate to yield

0

t

I I e τ−

= where LR

τ = .

Note: The inductor used in our experiment isn’t perfect. There is a resistance in the inductor also. All resistances in the above theory are total resistances for the circuit.

Overview of the experiment: We will use Data Studio to generate graphs of the time dependent voltages across the resistor and capacitor for an RC circuit and the time dependent voltages across the resistor and inductor for an RL circuit. These graphs created by Data Studio will be printed and used to answer several questions about the time dependent circuits. In addition, we will use some of the software functions to measure the time for the voltage to change by one half its total change.

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Experimental Setup and Procedure:

The RC circuit

Figure 2 shows the set up for the RC circuit.

Figure 2a. A picture of the circuit board wired for the RC time constant measurements. A jumper wire short circuits the inductor while the voltage probes are connected across the resistor and capacitor.

Figure 2b.

1. Turn on the computer and start Data Studio. Click on Create Experiment.

2. Select the voltage sensor and drag the icon to channel A. Repeat for channel B. Double click on one of the sensor icons to get the sensor properties window. Set the sample rate to 1000 Hz, Low (1X).

3. Click on the output (right side of 750 interface box image) and set up the following in the signal generator window: Positive Square Wave Function with an Amplitude of 4.0 V. Set the frequency (of the square wave) to 0.4 Hz. Close the window.

4. Select Sampling Options from the Experiment Setup window and set Automatic Stop to 4 seconds. Click on OK.

5. Wire the output of the Science Workshop interface to the circuit board as shown in Figure 2. Note that the interface has an output terminal and a ground terminal. The output terminal will be positive in this experiment. The 100 Ω resistor and 330 µF capacitor are used in this part of the experiment.

750 Interface Box

gnd.

jumper wire

Ch A + -

Ch B + - pos. sq. wave output

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6. Connect the wires of the voltage sensor from one channel to across the resistor and the wires of the voltage sensor from the second channel across the capacitor. Be sure to observe the polarity so the red wire is connected to the positive side of each element.

7. Set up to graph the data of both inputs from the voltage sensor on the same graph.

8. Click start to record the data.

9. Click on the Data button (right most button at the top of the graph) and deselect data grouping. Lock the origins of the horizontal axes. Enlarge the data portion of the graph appropriately and print the result. Label your graph as indicated in the analysis section. Maximize the graph window.

10. Turn off the Data Point Gravity by double clicking on the graph display, select Tools and set Data Point Gravity to 0.

11. In the Data window (upper left) double click on Voltage, ChA V). In the Data Properties window that opens select the Numeric tab and choose Scientific Notation. Set the number of digits to 3. Repeat the step for Channel B.

12. Click on the smart tool button located at the top of the graph. It looks like coordinate axes. Click and drag it to locate the coordinate for the time at the start of the charge cycle for the capacitor. (Hint: Enlarge portions of the graph as necessary so you can read the position accurately.) Record this time.

13. Move the cursor until you get a hand with a triangle near the index finger. Click and drag that icon to determine the change in time needed to obtain ½ the maximum voltage displayed. Use the fact that t1/2 = τ ln 2 to calculate the time constant. If it is difficult to get an accurate position for the time, you may want to enlarge your graph at this time.

14. Use the same click and drag process to determine t1/2 for the discharge portion of your graph.

15. Repeat steps 10 through 12 to measure the t1/2 times for the voltage across the resistor for both parts of the square wave cycle.

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The RL circuit.

Figure 3 shows the set up for the RC circuit.

Figure 3a. A picture of the circuit board wired for the RL time constant measurements.

Figure 3b.

1. Double click on one of the voltage sensor icons to get the sensor properties window. Change the sampling rate to 4000 Hz, Low (1X).

2. Double click on the output window, and set up the following in the signal generator window: Positive Square Wave with an Output of 4.0 V. Change the frequency (of the square wave) to 50 Hz.

3. Select Options from the menu and set auto stop to 0.05 second.

4. Wire the output of the Science Workshop interface to the circuit board as shown. Note that the interface has an output terminal and a ground terminal. The output terminal will be positive in this experiment. The 10 Ω resistor and 8.2 mH inductor are used in this part of the experiment. The inductor will have a resistance as well.

5. Connect the wires of the voltage sensor from one channel to across the resistor and the wires of the voltage sensor from the second channel across the inductor. Be sure to observe the polarity so the red wire is connected to the positive side of each element.

6. Set up to graph the data of both inputs from the voltage sensor on the same graph. Click on the Data button (right most button at the top of the graph) and deselect data grouping. Lock the origins of the horizontal axes. Enlarge the data

750 Interface Box

gnd.

Ch A + -

Ch B + - pos. sq. wave output

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portion of the graph appropriately and print the result. Label your graph as indicated in the analysis section. Maximize the graph window.

7. Turn off the Data Point Gravity. Set the data display to scientific notation with at least 3 digits.

8. Click start to record the data.

9. Maximize the graph window. Enlarge the data portion of the graph appropriately and print the result. Label your graph as indicated in the analysis section.

10. Click on the smart tool button located at the top of the graph. Locate the where the lines cross which show the coordinate values on your graph, click it and drag it to locate the coordinate for the time at the start of the positive voltage cycle for the resistor. Record this time.

11. Repeat the click and drag procedure to determine the maximum voltage across the resistor.

12. Repeat the click and drag procedure to determine the time when the voltage across the resistor was ½ of the maximum. Use the fact that t1/2 = τ ln 2 to calculate the time constant. If it is difficult to get an accurate position for the time, you may want to enlarge your graph at this time.

13. Use the same click and drag process to determine t1/2 for the decreasing resistor voltage portion of your graph.

14. Repeat the steps 9 through 11 for the inductor voltages. Note that the inductor has a resistance and will not have a zero voltage when the current reaches a steady positive value. You must determine the minimum voltage across the inductor and locate the time taken to determine ½ the change of inductor voltage for increasing current in the circuit. When the current is decreasing the voltage across the inductor will, of course, drop to zero. Again you want to determine the time for the inductor voltage to drop to ½ its maximum value.

15. To determine the resistance of the inductor, first determine the steady constant voltage across resistor R. Apply Ohm’s Law to calculate the current through the 10 Ω resistor. Nest divide this same current into the steady constant voltage across the inductor (Ohm’s Law again) to determine the inductor’s resistance.

Analysis and Report:

For the RC circuit:

1. Determine the average time constant from your measurements. How does this average compare to the time constant calculated using the values of R and C for the circuit elements you used?

2. On the printed graph of voltage vs. time, label the curve showing the voltage across the resistor, and the curve showing the voltage across the capacitor. Label the period of time corresponding to charging the capacitor, and the period of time corresponding to discharging the capacitor.

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3. What are the similarities and differences regarding the voltage across the resistor during each of these time periods?

For the RL circuit:

1. Determine the average time constant from your measurements. How does this average compare to the time constant calculated using the values of R and L for the circuit elements you used?

2. On the printed graph of voltage vs. time, label the curve showing the voltage across the resistor, and the curve showing the voltage across the inductor. Label the period of time corresponding to establishing a magnetic field in the inductor, and the period of time corresponding to discharging the magnetic field in the inductor.

3. What are the similarities and differences regarding the voltage across the resistor during each of these time periods?