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# RC and LR circuits Measuring the time constant

By Danny Marshall,2014-04-21 01:57
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RC and LR circuits Measuring the time constant

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 ? 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 ?

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

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

tt，~V0 and IeQCVe1！？：?0R；?

where t is time, I is the current in the circuit, Q is the charge on the capacitor, V is 0

the source voltage, R is the resistance, C is the capacitance and 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 ttVmax and QCVeIemaxR

LR Circuit

The LR circuit, shown schematically in Figure 1

behaves similarly. If we close the switch so the emfRvoltage source provides current to the circuit 1(position 1), Kirchhoff’s loop rule yields 2L VIR？？！E0sL

where V is the source voltage, I is the current in the 0

circuit, R is the resistance and E is the induced emf L

of the inductor. Substituting for the emf of the

inductor gives us

dIFigure 1. An RL circuit. When the VIRL？？！00switch is closed toward position 1, a dtcurrent will begin to flow through the We can solve this differential equation readily if we resistor and inductor. If, after a divide by R and then let a new variable x be defined current is established in the circuit, VVdxdIthe switch is quickly thrown to 00as . Since is a constant, and the I！？position 2, the magnetic field in the RRdtdtinductor will continue to push a differential equation becomes current through the circuit until the

energy in that field is exhausted. Ldx x；！0Rdt

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

xR ln！？txLo

LFirst we define , then we rewrite the above equation as an exponential equation R

to give us

tx exo

V0We note that I = 0 when t = 0, so . We then substitute the values for x and x xooR

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

t，~LV0 where = the time constant of the circuit. 1Ie！？：?RR；?

dIThe emf across the inductor is determined by E！？LLdt

tt？？，~，~VV1R00；； LeLeE！？？！？？：?：?LRRL；?；?

t EVeL0

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

dI or IR？！E0IRL；！0Ldt

We separate variables and integrate to yield

tL where . IIe0R

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.

Experimental Setup and Procedure:

The RC circuit

Figure 2 shows the set up for the RC circuit.

750 Interface Box

Ch A Ch B gndpos. sq. wave output + - + - . Ch A + -

jumper wire

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.

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, Fast.

3. Double click on the output window, 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 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

resistor and 330 (F capacitor are used in this part of the experiment.

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 clicking on the Data button (again) and

select the Graph Settings window. Set the Data Point Gravity to 0. 11. Click on the smart tool button located at the top of the graph. It looks like

coordinate axes. 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 charge cycle for the capacitor. (Hint: Enlarge portions of the

graph as necessary so you can read the position accurately.) Record this time. 12. Repeat the click and drag procedure to determine the time when the voltage

was ? of the maximum. Use the fact that t = ；;ln 2 to calculate the time 1/2

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 t for the discharge portion 1/2

14. Repeat steps 10 through 12 to measure the t times for the voltage across the 1/2

resistor for both parts of the square wave cycle.

The RL circuit.

Figure 3 shows the set up for the RC circuit.

750 Interface Box

Ch A Ch B gndpos. sq. wave output + - + - . Ch A + -

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, Fast.

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 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

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 by clicking on the Data button (again) and

select the Graph Settings window. Set the Data Point Gravity to 0. 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 ? of the maximum. Use the fact that t = ；;ln 2 to 1/2

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 t for the decreasing resistor 1/2

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 ? 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 ? 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;，？;， 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.

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?

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