LR Time Constants-SL F17V1 PDF

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RL Circuits Lab Lab Objectives   

Experimentally verify response of inductors as voltage is applied and removed (impulse response) Observe and model the differences between a real inductor component and an ideal inductor Verify the behavior of inductors in series

Materials   

 

Vernier voltage probe 9 or 36 mH Leyboldt Coil (the inductor) One key switch

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Two 10Ω resistors Vernier LabQuest® Mini and cable Computer

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Two D cell and holder Leads (alligator to banana and alligator to alligator)

Review- Theory of Inductance in RL Circuits – Response to Impulse Changes in Applied Voltage The relationship between the induced voltage and current in an inductor is derived from Faraday and Lenz Laws as1:

( Eq 1 ) V L=−L

dI dt

In this lab we will look at the time variation of the current through inductors in series with resistors as the inductor is energized and de-energized. This is analogous to looking at the voltage across capacitors as they are charged and discharged. Using Kirchhoff’s Voltage Law and Eq 1 above the current flow through an inductor in series with a resistor and the corresponding voltage across the resistor can be shown to be: De-Energizing the inductor– a voltage source is instantaneously removed from the circuit after being connected for a long time ( ≫ τ ¿ . −t

L ( Eq 2 a ) I =I 0 e τ where τ= ∧I 0=current at t=0 R −t

( Eq 2 b ) V R=IR=I 0 R e τ Energizing the inductor - a voltage ε 0 is applied at t=0

1−e ε −t L (¿ ¿ ) where I 0= 0 ∧τ= τ R R ( Eq 3 a ) I =I 0 ¿ 1 Subscripts indicate the circuit component. For examples V R = voltage across the resistor and through the inductor.

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I L =current

RL Circuits Lab

1−e −t (¿ ¿ ) τ ( Eq 3 b ) V R =IR=I 0 R ¿

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RL Circuits Lab Part 1 – Measuring the current as the inductor is energized The circuit below is used for this measurement. We are using the voltage probe to measure the voltage across the series resistor to determine the current through the circuit. Note that we could also use the current probe to directly measure I L . The current probe itself just uses a voltage measurement across a 0.1Ω internal resistor to measure the current. However it has components which introduce stray capacitances and inductances into the circuit which results in poorer results.

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Q1. What us the relationship between this voltage V R

and the current through the inductor I L ?

Q2 Sketch the expected time behavior of the current through the circuit and the voltages across the battery, inductor and resistor after the switch is closed at t=0. Q3 Explain the relative signs for V Battery ,V L and Lenz’s Law) in your prediction.

and

VR

(Use Kirchhoff Voltage Law

Q4 Given the values of the inductor and resistor calculate the expected value of the time constant τ for the inductor (coil) you have selected (either the 9 mH or 36 mH coil) The time constant is short compared to the times we usually measure with the Vernier probes. This requires a faster collection rate, here we will use, 10,000 samples per second. Construct the circuit as shown above. Use the two outer connections on the coil. On the LoggerPro® interface set the data collection (under the “Experiment” tab) as:

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RL Circuits Lab

Set the LoggerPro to show the V R versus time graphs. Start data collection and close the switch several times during the three second data collection period. You should see a graph similar to the following.

Rising Edges – Coil Energizing

Here we are interested in the current increasing as the coil is energized, the rising edge. Select one rising edge period and after appropriately rescaling the time axis, you should see a smooth curve for the resistor voltage as predicted by Eq 3b above. However, you may see what looks like one or several rapid on/off cycles due to switch “bounce”. If so repeat the process until a smooth curve is obtained.

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RL Circuits Lab

V Switch Bounce

t

Sample result with no “Bounce”

y= A ( 1−e

−Ct

Once a smooth curve is obtained use the LoggerPro

) +B

function to fit the curve.

Check the “Time Offset” option. Using the “Snipping Tool” insert the graph with the curve fit below.

Insert graph here Q5 What are the significance of the A, B and C parameters? How are they related to the time constant? Q6 What is the measured time constant

τ measured using the fit?

Q7 What is the percent error of τ measured versus the predicted value τ = causes for this difference?

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L ? What are possible R

RL Circuits Lab Note that the coil has a marked resistance . If we assume this is in series with the coil, we should add that into the resistance of the circuit as shown in the schematic below. (Sample shown for 9mH case. The 36mH would have a 9.5 internal resistance as marked.) Q8. Write an expression for the expected value of τ including this additional resistance.

Q9 Calculate the expected value and re-compute the percent error between the measured value and this new expected value. Q10 Do you obtain a smaller percent error? the magnitude of their contribution.

Identify factors which contribute to this error and estimate

De- energizing a coil: It may seem reasonable to construct a circuit analogous to that used to measure the discharge of a capacitor using the two position SPDT switch as shown below.

The above circuit will not work!

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RL Circuits Lab

However these switches break the contact with the battery before they make the contact completing the discharging LR circuit. For a short period of time there is an open circuit, R= ∞ in the equation

τ=

L . Therefore τ =0 in ths short time so the current rapidly collapses before the contact is R

made across the resistor and we never get to measure the LR time constant for the decay of the inductor current. The circuit below allows for the removal of the battery from the circuit without ever creating an open circuit.

10 Ω

Initially the switch is closed to energize the coil. The switch is opened and the voltage across the resistor is measured. This is proportional to the current through the inductor. Use the data collection setting as in the previous exercise. Construct this circuit. Start with the switch open. Start data collection and while collecting close the switch several times during the three second data collection period.

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RL Circuits Lab Sample results

Use an exponential fit to the data making certain to select the “Time Offset” option. −Ct

y= A e

+b

Q11 What are the values of A, B and C and what do they represent? What is the time constant? Q12 In the circuit shown what is the total resistance? What is the expected value of τ and the percent error between the measured and expected value? Include the internal resistance of the inductor in these calculations.

Inductors in Series and Parallel Design and perform experiment to show that inductors inductor

L1∧ L2 in series act as an equivalent

L12 = L1+ L2 when energizing two inductors in series.

Q13 Sketch the circuit including internal resistances of the coils. Q14 Identify and estimate the magnitude of significant sources of experimental error. For each indicate whether it is a systematic or random error. Q15 Describe the measurements and results. Using your data show if the time constant is consistent with L12 = L1+ L2 within the estimated value of the errors you indentify.

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RL Circuits Lab Design and perform experiment to show that inductors inductor

L12 =

L1∧ L2 in parallel act as an equivalent

1 1 1 when energizing two inductors in parallel. + L 1 L2

Q16 Sketch the circuit including internal resistances of the coils. Q17 Identify and estimate the magnitude of significant sources of experimental error. For each indicate whether it is a systematic or random error. Q18Describe the measurements and results. Using your data show if the time constant is consistent with

L12 =

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1 1 1 + L 1 L2

within the estimated value of the errors you indentify.

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