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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

    of your graph.

    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

    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 ? 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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