RL Circuit

Equipment Capstone with 850 interface, 2 voltage sensors, RLC circuit board, 2 male tomale banana leads approximately 80 cm in length

1 Introduction

The three basic linear circuit elements are the resistor, the capacitor, and the inductor. Thislab is concerned with the characteristics of inductors and circuits consisting of a resistor andan inductor in series (RL circuits). The primary focus will be on the response of an RLcircuit to a step voltage and a voltage square wave.

2 Theory

2.1 Inductors

An inductor is a 2-terminal circuit element that stores energy in its magnetic field. Inductorsare usually constructed by winding a coil with wire. To increase the magnetic field inductorsused for low frequencies often have the inside of the coil filled with magnetic material. (Athigh frequencies such coils can be too lossy.)Why? Inductors are the least perfect of thebasic circuit elements due to the resistance of the wire they are made from. Often thisresistance is not negligible, which will become apparent when the voltages and currents inan actual circuit are measured.

If a current I is flowing through an inductor, the voltage VL across the inductor isproportional to the time rate of change of I, or dI

dt. We may write

VL = LdI

dt, (1)

where L is the inductance in henries (H). The inductance depends on the number of turns ofthe coil, the configuration of the coil, and the material that fills the coil. A henry is a largeunit of inductance. More common units are the mH and the µH. A steady current througha perfect inductor (no resistance) will not produce a voltage across the inductor. The signof the voltage across an inductor depends on the sign of dI/dt and not on the sign of thecurrent. Figure 1 shows the relationship between the current and voltage for a resistor and ininductor. The arrow indicates the positive direction of the current through the two devices(resistor and inductor). That is, if the current is in the direction of the arrow, then I > 0.[Note that just because the arrows point down in the Fig. 1 doesn’t mean that the current is

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Intro Experimental Physics II Lab: RL Circuit

Figure 1: Relationship between current and voltage in a resistor and inductor.

flowing in the downward direction. The current can also flow in the upwards direction, butin that case I < 0.] VR and VL are the potential differences across the resistor and inductor,respectively, defined such that if V(R,L) > 0, then the potential at the “+′′ end of the deviceis greater than the potential at the “−′′ end. [According to this convention, if V(R,L) < 0then the potential at the bottom is greater than the potential at the top.]

According to these conventions, for the case of the resistor, V = IR. If the current I > 0(current flowing downward) the potential is greater at the top than at the bottom in Fig. 1.In the case of the inductor, if dI/dt > 0 then the potential is greater at the top of theinductor than at the bottom. In the inductor, one can have the situation where the currentis flowing downward (I > 0), but decreasing in magnitude so that dI/dt < 0. In this caseVL = dI/dt < 0, so the potential at the top is smaller than at the bottom. The potentialis always lower on the “downstream” end of a resistor, but for an inductor, itcan be larger or smaller, depending on the rate of change of the current.

As mentioned above, the coil of wire making up an inductor has some resistance RL.This resistance acts like a resistor RL that is in series with an ideal inductor. (An idealinductor is one with no resistance.) In this situation, the voltage across the inductor is

VL = LdI

dt+RL I. (2)

2.2 RL Circuits

A series RL circuit with a voltage source V (t) connected across it is shown in Fig. 2. Thevoltage across the resistor and inductor are designated by VR and VL, and the current aroundthe loop by I. The signs are chosen in the conventional way; I is positive if it is in thedirection of the arrow. Also, the positive signs of the voltages across the various componentsare indicated by the + and − symbols. Kirchoff’s law, which says that sum of the voltagechanges around the loop is zero, may be written

VL + VR = V, (3)

where it is assumed that the voltages V (t), VR(t), and VL(t) can all vary with time t.Assuming RL = 0 (any resistance in the inductor can just be added to that of the resistor)and letting VL = LdI/dt and VR = IR, Equation (3) becomes

LdI

dt+RI = V. (4)

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Intro Experimental Physics II Lab: RL Circuit

Figure 2: RL circuit: The loopy arrow indicates the positive direction of the current. The +and − signs indicate the positive values of the potential differences across the components.

The solution to the homogeneous equation [V (t) = 0] is I(t) = I0e− t

L/R , where I0 is the

current through the circuit at time t = 0. This solution leads immediately to VR = I0Re− t

L/R

and VL = −I0Re−t

L/R . The homogeneous solution decays exponentially with a time constantτ = L/R.

Of interest is the response of an RL circuit to an abrupt change in the voltage V of thesource from V = 0 to a constant voltage V = V0. Before the change, we assume that the

voltage of the source was at V = 0 for a long time (much longer than τ). The function e−t

L/R

describes the time dependence of the circuit. To find the behavior as a function of time, itis only necessary to use physical intuition and put the appropriate two constants into thesolution. One guiding principle is that the current I, and therefore VR does not change theinstant the voltage across an RL circuit is changed. That is, VR and I are continuous. Thismeans initially, that the entire voltage change must appear across the inductor.

QUESTION: What is the basis for the argument that I doesn’t change discontinuouslywhen the source voltage is suddenly changed from V = 0 to V = V0?

A solution to the inhomogeneous equation

LdI

dt+RI = V0 (5)

is I = V0/R. The general solution in this case is a linear combination of the homogenousand inhomogeneous solutions that satisfy the initial and final conditions. This is

I =V0R

(1 − e−

tL/R

). (6)

This says that after the constant voltage V0 is applied, the current in the circuit will expo-nentially approach V0/R if the inductor has no resistance, or V0/(R + RL) if the inductorhas a resistance RL. These considerations apply whatever the previous history of the RL

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Intro Experimental Physics II Lab: RL Circuit

VR Voltage VR across the resistor

0

V0

time

VL Voltage VL across the inductor

0

0

V0

Figure 3: Voltages VR across the resistor R and VL across the inductor L as a function oftime after the source voltage is switched from V = 0 to V = V0.

circuit. Consider an RL circuit where V = 0 and there is no current. We assume that RL =0. If at t = 0 a constant voltage V0 is put across the circuit, for t ≥ 0, VL and VR are givenby,

VL = V0 e− t

L/R and VR = V0

(1 − e−

tL/R

). (7)

This behavior is illustrated in Fig.3. The current is not plotted, but remember that thecurrent is proportional to VR. Initially all of V0 appears across L because the current justbefore and after the application of V0 is zero. As the current exponentially builds up thevoltage across the resistor increases and the voltage across the inductor decreases. If we waitmany time constants τ = L/R, the voltages reach steady state, and we have VR ∼= V0 andVL ∼= 0. If now V is set equal to 0 (this is equivalent to shorting the circuit) VL and VR willbe given by

VL = −V e−t

L/R and VR = V e−t

L/R . (8)

The response of an RL circuit to an alternating pair of constant voltages, first at V0, and then0, can be observed by applying a square wave to the circuit, alternating between V = V0 andV = 0. Figure 3 shows half the period of such an oscillation for the case where the period Tis much larger than the time constant τ = L/R.

QUESTION: How would Fig. 3 be modified if the inductor had some resistance?

QUESTION: If T L/R, what is the response of an RL circuit to a symmetric squarewave that oscillates between +V and −V (assume RL = 0)?

If a high frequency square wave, such that T L/R, is applied to an RL circuit, the

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Intro Experimental Physics II Lab: RL Circuit

changes in current and VR are minimal. There is not enough time for the current through theinductor to change much before the voltage is reversed. If the square wave is not symmetricwith respect to ground the average VR will be the average voltage of the square wave,assuming RL = 0. Fig. 6 shows the voltages for a square wave that oscillates betweenthe constant voltage V and 0 (ground). The exponential dependences of VL and VR areapproximated as straight lines and it has been assumed that RL = 0 and T L/R.

3 Procedure

3.1 Measuring RL

In this section the DC value of RL will be measured.

• Run the Capstone software.

• Go to the hardware setup window.

• Click on the output of the 850 interface and select Output Voltage - Current Senor.

• In the tools column, select Signal Generator icon and click on the 850 output 1.

• In waveform, pick DC, program it for 2 volts DC, and click on Auto.

• Next, setup a digits display and in select measurements pick Output Current.

• Program the digits display for at least 2 decimal places.

• Connect the output of the 850 interface directly across the coil (see Fig. 4)

• Click Record and then Stop.

From your data calculate RL, the resistance of the coil. Click the FILE menu button andthen New Experiment. Remove the wires from the coil.

During your experiments compare the value of RL to the values of the resistances usedin the RL circuits.

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Intro Experimental Physics II Lab: RL Circuit

Inductor

InductorCore

Resistors

Figure 4: Picture of the circuit board

3.2 Setting up the RL circuit and using Capstone

In this section, the response of an RL circuit will be examined experimentally using thesignal generator, 2 voltage sensors, and the oscilloscope display.

• In Capstone, go to the Hardware Setup window. Click on the output of the 850interface and select Output Voltage - Current Senor.

• Program the analog inputs of the 850 interface for the two voltage sensors. Next, dragthe scope icon to the center of the white screen.

• Resize the windows by using the orange tack. Add two additional y axis’s to the scopeby clicking the Add new y-axis to scope display icon twice.

• On the scope display, click Select Measurement on each axes, and set one as OutputVoltage, and the other two as the analog voltages.

In the experiment, various resistors will be connected in series with an 8.2 mH inductor.Hook the circuit and voltage sensors up as shown in Fig. 7, paying attention to the polaritiesof the leads and terminals. The polarities are very important in this experiment. Apositive voltage from a voltage sensor will correspond to a positive voltage as given by theconvention of Fig. 2. The input square wave to the RL circuit will be in the Output Voltagechannel, VL and VR will be on the other two channels.

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Intro Experimental Physics II Lab: RL Circuit

You may wonder about the particular choice of frequencies used in the experiments. Itturns out that the results are better if the signal generator does not operate at too high afrequency and the time constants and frequencies have been chosen accordingly.

Remarks on running Capstone:

• When using the oscilloscope, use Fast Monitor Mode rather than Continous Mode.

• You need to use the trigger function on the scope to obtain a stable signal. Click onthe trigger icon and on the side of the scope move the trigger arrow up.

• For reliable triggering with a positive only square wave, make the trigger voltage pos-itive but less than 1 V.

• The output of the 850 interface will be controlled by the 850 output 1 signal generator.Remember you need to click on Auto!

The items in red will be the functions you will be using and adjusting when you runthe experiment.

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Intro Experimental Physics II Lab: RL Circuit

• Do not apply a voltage amplitude of greater then 5 V from the signalgenerator! Use no more then 3 V. (The current rating for the inductor is 0.8 A.)

• When you click STOP, the last traces are stored. The stored traces are better forexamination than the “live” traces as they are steady.

• You can use the Show data coordinates and access delta tool on the stored traces.

• On the stored traces you can use many of the scope functions.

• You can print out the stored traces.

• Label your printed out traces immediately as V, VL, and VR as the printers are notcolor printers.

• Use the same vertical sensitivity for all 3 traces so that you can easily compare thetrace values.

• Due to RL, your graphs will differ somewhat from Figs. 3, 5, and 6 of this write-up,which assumes RL = 0.

4 Experiments

4.0.1 T L/R, Positive Only Square Wave

Hook up an RL circuit with the R = 33 Ω resistor. Compute the expected value of the timeconstant L/Rtot (with uncertainty) for two cases: RL = 0 and RL given by your result fromthe previous section. How do these values compare with the period (T ) of a 300Hz square wave?

• Examine VL, VR, and V using a 300 Hz square wave that oscillates between ground (0V) and 2V.

• Be sure to set the sample rate to 5000 Hz. Are the traces what you expect?

• Now to determine the time constant from the data. We want to plot one “charging”cycle in Python.

• In Capstone create a table recording VL and V .

Write down the sampling rate, RECORD data for a very short time (less than one second).In order to record data, it is necessary to close the scope display, then click start andstop almost immediately. Click File, Export Data and choose your desired run from thevoltage sensor which is across the conductor. Save the file. Repeat for the data of Output

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Intro Experimental Physics II Lab: RL Circuit

Voltage. Open the data file in a text editor and look for one “chunk” of data with V ≈ 2V.Copy these lines into a new file, this will be the data used for plotting.

We expect VL = V e−t

R/L . Using your data, make a plot of log(VL) versus t in Python,and determine the slope and y-intercept (with uncertainties). What do you expect theslope to be? The y-intercept? From your slope and intercept, compute the timeconstant and V . Do they agree with the expected values? Which expression forthe time constant is best to use: L/R or L/(R +RL)?

4.0.2 T L/R, Positive Only Square Wave, Metal Core

Use the same setup as the previous section. With the Capstone scope running, insert themetal rod into the inductor coil. How has the trace changed? What parameter isthe metal rod changing?

With the rod inside the coil, redo the analysis from the previous section. After closingthe scope display, record a table of VL and V for a short time, export the table, and copyout one section of data with V ≈ 2V. Plot log(VL) versus t (the sampling rate should bethe same), and determine the slope and intercept. Are the numbers the same as in theprevious section?

Use the slope to calculate the new inductance L′ ± δL′, assuming the resistance Rtot isthe same as in the previous section. How significant is the effect of the metal core?

4.0.3 T L/R, Symmetric Square Wave

Use the same parameters as in section 4.0.1 except use a square wave that is symmetric withrespect to ground (varies from 2V to -2V).

The time constant τ = L/Rtot is defined to be the time it takes VL to reduce by a factorof e (=2.718...). Knowing this, we can approximate the time constant visually with theShow Data Coordinates. In Capstone, click RECORD then click STOP to freeze thetrace. Find the peak of the VL curve, and record the voltage Vpeak and the time tpeak. If weassume RL = 0, then VL should decay to 0 with time. Using Show Data Coordinates,find the point on the VL curve where VL = Vmax/e, remember 1/e ≈ 0.37. Record the timeof this point t1. Then τ = t1 − tpeak, simple!

Compare your results to what you expect. Due to limitations in the equipment used andthe non-zero value of RL, the peak voltages will be less than calculated.

4.0.4 T L/R, Positive Only Square Wave

Apply a 3,000 Hz positive only square wave to an RL circuit with R = 10 Ω. Change thex-axis interval to .5 ms/div and set the sample rate to 50,000 Hz. What is the expectedtime constant and how does it compare to T? Compare your results to Fig. 3.In Fig. 3, is approximating the exponentials by straight lines reasonable? Whyis there distortion in the square wave?

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Intro Experimental Physics II Lab: RL Circuit

4.0.5 T L/R, Symmetric Square Wave

Use the same parameters as in section 4.0.4 except use a square wave that is symmetric withrespect to ground. Compare to Fig. 4. The effect of RL is quite small here as the averagecurrent is small.

4.0.6 T ≈ L/R, Symmetric Square Wave

Apply a 500 Hz square wave to an RL circuit with R = 10 Ω. Are the results what youexpect? What occurs when you try lower and higher frequencies?

4.0.7 RLC Circuit (Optional)

Make a series circuit with the 33 Ω resistor, the inductor, and the 330 µF capacitor. This iscalled an RLC circuit, for obvious reasons. Use the voltage sensors to monitor the voltageacross the inductor and the capacitor (we will ignore the resistor). Monitor the voltages inthe Capstone oscilloscope, and apply a 300 Hz, 2V, positive-only square wave to the circuit.Change the sample rate to 5000 Hz. Describe what you see.

5 Finishing Up

Please leave the lab bench as you found it. Thank you.

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Intro Experimental Physics II Lab: RL Circuit

Figure 5: The signal in the first row is the voltage across the inductor. The signal in thesecond row is the voltage across the resistor and the bottom row is the voltage supply.

Figure 6: The signal in the first row is the voltage across the inductor. The signal in thesecond row is the voltage across the resistor and the bottom row is the voltage supply.

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