Demonstrations with an LCR CircuitYaakov Kraftmakher, Bar-Ilan University, Ramat-Gan 52900, Israel

The LCR circuit is an important topic in the course ofelectricity and magnetism. Papers in this field con-sider mainly the forced oscillations and resonance.l >

Our aim is to show how to demonstrate the free and self-excited oscillations in an LCR circuit.

The setupWith an inexpensive oscilloscope, it is easy to demonstrate

free oscillations in a series LCR circuit. We use a Kenwood CS-4025 model oscilloscope (Fig. 1). The LCR circuit consists ofan SOO-turn coil from PASCO scientific (SF-S611), a 100-nFcapacitor, and a variable 100-0 resistor; instead of the latter,several fixed resistors can be used. The inductance of the coilis 12ml-l. The capacitor C is connected to the Y- input of theoscilloscope. Rectangular electric pulses from the oscilloscopeintended for calibration purposes (100/120 Hz, 1V peak-to-peak) are used to initiate free oscillations. The source of thepulses is connected to the LCR circuit through a capacitor C1 =

1 nP. The capacitor C1 and the LCR circuit form a differentiat-ing circuit, so short positive and negative electric pulses areperiodically applied to the capacitor C [Fig. 2(a)]. The imped-ance of the calibration output is much lower than the loadimpedance.

Free oscillationsThe first demonstration shows changes in the frequency

and decay of the free oscillations caused by changing the pa-rameters C and R. The frequency of undamped oscillationsin an LC circuit is Wo = (LCt'h.. Due to energy losses in thecircuit, the frequency of free oscillations w becomes lower: w2

= w02 - "(2, where "(= RI2L is called the decay constant. Herethe resistance R reflects all sources of the energy losses in thecircuit: the resistance of the coil L and of the resistor added,

Fig. 1. Photo of the setup.

168 THE PHYSICS TEACHER. Vol. 49, MARCH 2011

losses in the capacitor, resistance of the source of rectangularpulses, input resistance of the oscilloscope input, and lossesfor radiation of electromagnetic waves. This resistance is notshown in Fig. 2. In our case, y «W02, so w is close to w00 Theoscillations decay exponentially; that is, the amplitudes of thevoltage on the capacitor C and of the current in the circuit areproportional to exp( -"(t). The free oscillations are observedon the oscilloscope screen for several values of C and of thevariable 100-0 resistor.

Using magnetic or conducting coreIn the second demonstration, a ferrite or an aluminum

rod is put inside the inductor L. With the ferrite rod, the fre-quency of free oscillations decreases, while the decay of theoscillations becomes slower (Fig. 3). Due to the high magneticpermeability of the ferrite, the magnetic flux through the coilincreases, so the inductance of the coil also increases.

With the aluminum rod, IS mm in diameter, the frequencyof free oscillations increases due to the decrease of the induc-tance of the coil. When a conductor is subjected to an externalac magnetic field, eddy currents appear in it (Faraday's law).These currents tend to reduce the magnetic field inside thecoil (Lenz's law). Therefore, the magnetic flux through the coildecreases, so the inductance of the coil also decreases. Themagnetic field inside the conductor depends on the frequencyof the magnetic field and on the resistivity of the conductor.This phenomenon is called the skin effect and can be used forcontactless measurements of resistivity of nonmagnetic con-ductors6,7 and for metal detection.f With the aluminum rodinside the coil, the decay constant increases. An additionalenergy loss arises due to the eddy currents in the rod. For theferrite rod, this effect is negligible due to the high resistivity ofthe ferrite.

output of r------,channel1D

(a) (b) (c)

Fig. 2. Diagram of the setups: (a) basic LCR circuit; (b) LCR circuitwith feedback; (c) circuit for observing self-excited oscillations.The resistance reflecting all energy losses in the circuit is notshown.

001: 10.1119/1.3555505

Fig. 3. Examples of free oscil-lations: (a) basic LCR cir-cuit; (b) ferrite rod inside theinductor; (c) aluminum rodinside the inductor.

The feedback

Fig. 4. Examples of oscilla-tions with feedback: (a) posi-tive feedback; (b) negativefeedback; (c) self-excitedoscillations.

The third experiment demonstrates positive or negativefeedback [Fig. 2(b)]. Now there is no need for the 100-Sl vari-able resistor. The oscilloscope has an output terminal fromone amplification channel (channel l ) and serves also as anamplifier. The gain of the channel depends on the arrangedsensitivity. The maximum gain achieved with the sensitiv-ity of 1mVIdiv is 100; for 0.1 VIdiv, the gain becomes unity.The voltage across the capacitor C is amplified and then fed,through a variable resistor R [ = 5 kSl, to a coil L[ (feedbackcoil) inductively coupled to the coil L. As the variable resistor,we use a PASCO decade resistance box (PI-9588). A 400-turncoil (PASCO SF-861O) serves as the feedback coil and is putadjacent to the coil L. The current in the feedback coil L[ in-duces an emf in the coil L. It is easy to see that this emf is inphase with or opposite to the current in the LCR circuit, thusproviding positive or negative feedback. The sign of the feed-back depends on the polarity of the two coils. The feedbackenhances or reduces the decay constant of the free oscilla-tions, while the oscillation frequency remains the same (Fig.4). Therefore, it is easy to distinguish the changes in the decaycaused by the feedback and those due to the ferrite or alumi-num rod inside the coil of the LCR circuit.

Self-excited oscillations

The strength of the feedback depends on the distancebetween the coils Land L t- on the amplification of the oscil-loscope, and on the variable resistance R[. With positivefeedback, continuous self-excited oscillations are achievable.For this demonstration, the calibration output is disconnectedfrom the LCR circuit [Fig. 2(c)].

For creating self-excited oscillations, the energy suppliedto the LCR circuit by the feedback should outweigh the energylosses. In this case, the decay constant, becomes negative, sothe oscillation amplitude should exponentially and unlimit-edly increase. Clearly, this is impossible. When the oscillationamplitude increases, the amplification in the feedback circuitunavoidably decreases because the output signal of the cir-cuit is evidently limited. It is easy to demonstrate this effectby positioning a pair of semiconductor diodes parallel to theoutput of the oscilloscope. The diodes are put in parallel butwith opposite polarities. For high signals, the resistance of thediodes decreases and their shunting action becomes stronger;therefore, the signal in the feedback circuit decreases.

When the self-excited oscillations are achieved, their am-plitude becomes out of scale of the oscilloscope. To see theoscillations nevertheless, we use channel 2 of the oscilloscopewith properly set sensitivity. The capacitor C is connected toboth channels: channell provides the feedback, while chan-nel2 serves for observing the voltage across the capacitor.

Intentionally, the inductor L and capacitor C were cho-sen to obtain the resonant frequency of the LCR circuit inthe audio range. To hear the oscillations, an audio amplifierwith a loudspeaker is connected to the output of the chan-nell. With the PASCO 800- turn coil and C = 100 nF, theresonant frequency is nearly 4.5 kHz. With the ferrite insidethe coil, it becomes nearly 2.5 kHz, and with the aluminumrod nearly 5 kHz. The changes in the frequency thus becomeaudible for the audience. Since the rods inside the coil L leadto changes in the decay constant, parameters of the feedbackcircuit should be readjusted for maintaining the self-excitedoscillations. Instead of the variable resistor R i- a suitable light-dependent resistor (LDR) can be put in the feedback circuit.In this case, the continuous oscillations are initiated by illumi-nating the LDR.

The experiments provide good opportunity to considersome additional subjects (the magnetic permeability, the skineffect, the feedback, the light-dependent resistor). Also, theyseem to be attractive topics for student projects. The self-os-cillation amplitude strongly depends on the distance betweenthe coils Land Ll (or on the position of a magnetic core insidethe coils) or on the resistance of the LDR in the feedback cir-cuit. This feature can be used for designing sensors of positionor illuminance.

Many thanks to the referee for useful suggestions.

THE PHYSICS TEACHER. Vol. 49. MARCH 2011 169

References1. G. Chadha, "Using an X-Y plotter to illustrate electrical reso-

nance in an LCR series circuit;' Phys. Educ. 21, 187-188 (1986).2. V.Ramachandran, "Revisiting the LCR circuit;' Phys. Educ. 26,

318-321 (1991).3. Se-yuen Mak, "The RLC circuit and the determination of in-

ductance;' Phys. Educ. 29,94-97 (1994).4. Philip Backman, Chester Murley, and P. J. Williams, "The

driven RLC circuit experiment;' Phys. Teach. 37,424-425 (Oct.1999).

5. Robert A. Morse, "Conceptualizing ideal circuit elements in theAP Physics C syllabus;' Phys. Teach. 43, 540-543 (Dee. 2005).

6. Yaakov Kraftmakher, "Eddy currents: Contactless measure-ment of electrical resistivity;' Am. f. Phys. 68,375-379 (April2000).

7. Yaakov Kraftmakher, "Contactless measurement of resistivitywith Data Studio,"Am. f. Phys. 77, 953-955 (Oct. 2009).

8. Yaakov Kraftmakher, "Metal detection and the Theremin in theclassroom;' Phys. Educ. 40,167-171 (2005).

Yaakov Kraftmakher is the author of more than 150 scientific papersand of three books: Lecture Notes on Equilibrium Point Defects in Metals(World Scientific, 2000), Modulation Calorimetry: Theory and Applications(Springer, 2004), and Experiments and Demonstrations in Physics: Bar-lianPhysics Laboratory (World Scientific, 2007).

Department of Physics, Bar-lian University, Ramat-Gan 52900, Israel;krafiy@mail.biu.ac.iI

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The LCR Circuit with Battery modelsimulates a resistor R, a capacitor C, and an inductorL in series with a battery and plots the time depen-dence of the voltage drops across these elements.Users can vary the circuit component values, cantoggle the battery voltage Vs between two values,and can change the time resolution t:,.t.

www.compadre.org/OSP/items/detail.dm?ID=10586

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.101:.- --"- __ -"- __ ~ ~_---.JRelated items can be found in the OSP Collection bysearching for "circuit" In particular, the LCR Circuitwith Function Generator model explores an LCR cir-cuit in series with a sinusoidal or square-wave function generator.

www.compadre.org/osp/items/detail.dm?ID=9469

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These supplemental simulations for the preceding paper by Yaakov Kraftmakher have been approved by theauthor and the TPT editor. Partial funding for the development of this model was obtained through NSF grantDUE-0937731.

THE PHYSICS TEACHER. Vol. 49, MARCH 2011

Wolfgang Christian

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