The RLC system: An invaluable test bench for studentsPierre Cafarelli, Jean-Philippe Champeaux, Martine Sence, and Nicolas Roy Citation: Am. J. Phys. 80, 789 (2012); doi: 10.1119/1.4726205 View online: http://dx.doi.org/10.1119/1.4726205 View Table of Contents: http://ajp.aapt.org/resource/1/AJPIAS/v80/i9 Published by the American Association of Physics Teachers Related ArticlesMotion tracking in undergraduate physics laboratories with the Wii remote Am. J. Phys. 80, 351 (2012) Understanding Pound-Drever-Hall locking using voltage controlled radio-frequency oscillators: An undergraduateexperiment Am. J. Phys. 80, 232 (2012) Vernier scales and other early devices for precise measurement Am. J. Phys. 79, 368 (2011) Robots in the introductory physics laboratory Am. J. Phys. 76, 895 (2008) Magnetometer construction and applications for introductory physics Am. J. Phys. 76, 807 (2008) Additional information on Am. J. Phys.Journal Homepage: http://ajp.aapt.org/ Journal Information: http://ajp.aapt.org/about/about_the_journal Top downloads: http://ajp.aapt.org/most_downloaded Information for Authors: http://ajp.dickinson.edu/Contributors/contGenInfo.html

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The RLC system: An invaluable test bench for students

Pierre Cafarelli and Jean-Philippe ChampeauxUniversite Paul Sabatier, IRSAMC, LCAR, CNRS UMR 5589, 118 Route de Narbonne,31062 Toulouse Cedex 4, France

Martine SenceUniversite de Toulouse, UPS, IRSAMC, LCAR, CNRS UMR 5589, 118 Route de Narbonne,31062 Toulouse, France

Nicolas RoyTop Modal, Ecoparc II, Avenue Jose Cabanis 31300 Quint-Fonsegrives, France

(Received 9 September 2011; accepted 22 May 2012)

Future scientists and engineers must master fundamental notions of instrumentation. A lab session

milestone based on the classical RLC circuit is proposed to check students’ theoretical and

practical knowledge. Topics include impedance, signal and linear system analysis in the time and

frequency domains, as well as the fundamental relations connecting them. The numerical

implementation of the Fourier transform and its related pathologies (aliasing and leakage) are also

discussed. Practical skills are evaluated through the intensive use of instruments and by presenting

and solving potential problems that could alter the accuracy of measurement. VC 2012 American

Association of Physics Teachers.

[http://dx.doi.org/10.1119/1.4726205]

I. INTRODUCTION

Instrumentation is a synergy of hardware and softwareresources used to extract quantitative information fromunknown signals or systems. Obtaining reliable measurementswith their uncertainties1,2 requires a global and sometimesdetailed knowledge of each resource, the search for influentialparameters, and a final data analysis. This is a difficult andtime-consuming task requiring experience and know-how. Thelab session described in this article allows students to revisit thebasic skills required with the help of a causal linear and timeinvariant system (LTIS): the simple and familiar RLC circuit.

After a brief summary of LTIS mathematics,3,4 the RLCsystem and its associated instrumentation are presented.Experiments dedicated to system identification and valida-tion of LTIS fundamental relations are described, com-mented, and analyzed. This lab session improves thestudent’s skills in the use of standard instruments such asgenerators, oscilloscopes, and data acquisition cards (DAQ).It also reviews general concepts, offers numerical implemen-tation of fundamental relations, and pays particular attentionto sources of error (measurements and protocols). Thanks toready-to-use applications, no programming knowledge isrequired for DAQ control, so students can focus on settings.

Since the RLC system is mathematically identical to theubiquitous and fundamental damped spring-mass system,5 itprovides a low cost, simple, and excellent means to illustratethe basics of modal analysis (theory6 and testing7).

II. THEORETICAL BACKGROUND

Let us consider a system for which the input signal IN rep-resents the excitation and the output signal OUT representsthe response. In the time domain (subscript T), these signalsare described by time dependent functions.

We note first that INT(t) can be represented by a contin-uum of impulses

INTðtÞ ¼ðþ1�1

INTðsÞdðt� sÞds ¼ INTðtÞ � dðtÞ; (1a)

where � represents the convolution operator. Therefore, cau-sality, linearity, and time invariance imply

OUTTðtÞ¼ðt

�1INTðsÞhTðt�sÞds¼INTðtÞ�hTðtÞ: (1b)

This well-known result8 states that the output signal is givenby the convolution of the input signal with the impulseresponse hT(t).

Defining the Fourier transform of a function of time XT(t)by XFðxÞ ¼

Ðþ1�1 XTðtÞe�jxtdt and applying it to Eq. (1b)

gives a very simple relation in the frequency domain

OUTFðxÞ ¼ INFðxÞ:hFðxÞ; (2)

where hF(x), also denoted H(x), is called the transferfunction.

Another useful relation defines the correlation between theinput and the output signals

CIN;OUTðtÞ ¼ðþ1�1

INTðtþ sÞOUTTðsÞds: (3a)

For a LTIS, the integral is equivalent to

CIN;OUTðtÞ ¼ CIN;INðtÞ � hTðtÞ; (3b)

stating that cross-correlation between the output and inputsignals is given by the convolution of the input signal auto-correlation with the impulse response.

The above relations are used when one has to deal withone of the three issues involving LTIS: system identification,signal processing, and data deconvolution (see Fig. 1). Iden-tification refers to the process of deducing a mathematicalmodel of the internal dynamics of a “black-box” systemfrom observations of its output and input. Signal processinginvolves computation of the output signal, and finally datadeconvolution refers to the recovery of the excitation signalfrom the output measurement and system characteristics.

Usually numerical methods are used to compute or imple-ment the above relations. The most important tool is the FFT

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(fast Fourier transform), an efficient algorithm used to evalu-ate the discrete Fourier transform,

~XFðkxs=NÞ ¼XN�1

n¼0

XTðnTsÞe�j2pkn=N

( )k¼0;1:::;N�1

; (4)

considered to be a good estimate of XF(x). This relation usesN values of XT sampled at a frequency of fs ¼ xs=2p ¼ 1=Ts.

III. EXPERIMENTS

The RLC experiment uses the two-port network displayedin Fig. 2, composed of a resistor, an inductor, and a polymercapacitor. Each dipole is labelled with the value of the com-ponent, whose type is determined automatically by a PhilipsPM6303 RLC-meter. The following values were obtained bycombining the measurements given by this instrument andanother RLC-meter (Metrix IX3131) operating at the samefrequency (1 kHz): R¼ 100.0 X(60.3%); L¼ 17.4 mH(63%); and C¼ 9.797 nF (60.1%).

To perform this experiment, one needs a voltage followerwith a symmetric supply (þ15 V/�15 V), an envelope detec-tor (diode demodulator), and the following devices: an arbi-trary waveform generator (Agilent 33220A), an oscilloscope(Agilent 6012A), and a DAQ (National Instruments PCI6251). The DAQ is connected to a PC running Windows XPand controlled with LABVIEW 7.1 (National Instruments). Alow-noise preamplifier (Stanford Research SR560) andanother generator (Agilent 33521A) were used to performcomplementary experiments.

A. System identification

In general, system identification proceeds in two overlap-ping phases: structural identification in which the form of thelinear differential equation is determined, followed by pa-rameter estimation in which values for its coefficients areobtained. Here, structural identification is explicitly deduced

from the circuit displayed in Fig. 2. Circuit analysis providesthe well-known equation

LCd2OUTT

dt2þ RC

dOUTT

dtþ OUTTðtÞ ¼ INTðtÞ; (5a)

with the usual derived parameters

xn ¼1ffiffiffiffiffiffiLCp ðnatural frequencyÞ;

b ¼ R

2xnL¼ RCxn

2¼ R

ffiffiffiffiCp

2ffiffiffiLp ðdamping factorÞ;

Q ¼ Lxn

R¼ 1

2bðquality factorÞ:

(5b)

The spring-mass system study shows a formal analogywith this electrical system. The mass is analogous to L, thestiffness to 1/C, and the viscous damping element to R. Fur-thermore, the input voltage IN divided by C corresponds tothe force applied on the mass, and the output voltage OUT(across the capacitor) is equivalent to the mass displacement.Identification of both second order systems involves charac-terization of b (or Q) and xn. From an experimental point ofview, these parameters are usually deduced from the analysisof three other quantities. The first is the response to a voltagestep E0U(t), where U(t) is the Heaviside function

STðtÞ¼E0UðtÞ 1�e�bxntcosðxptÞ�bxn

xpe�bxntsinðxptÞ

� �;

(6)

with xp ¼ xn

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� b2

p. The second is the Green function or

response to the Dirac pulse,

hTðtÞ ¼ E0xpe�bxntUðtÞ 1þ bxn

xp

� �2" #

sinðxptÞ: (7)

And the third is the transfer function,

HðxÞ ¼ 1

1� xxn

� �2

þ j1

Q

xxn

: (8)

If b is less thanffiffiffi2p

=2, then |H(x)| goes through a maximum

Hmax at xr ¼ xn

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 2b2

p(the resonant frequency) whose

width (half-power method) is Dxr. Elementary algebra gives

Hmax ¼1

2b1ffiffiffiffiffiffiffiffiffiffiffiffiffi

1� b2p (9)

and

QC ¼xr

Dxr

� ��3dB

¼ffiffiffiffiffiffim2pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

m2 þ 2bffiffiffiffiffiffim1pp

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffim2 � 2b

ffiffiffiffiffiffim1pp ; (10)

with mk ¼ 1� kb2.With the previously defined values, the above parameters

are estimated (see Table I).The accuracy of a given parameter W(X,Y,..) is calculated

according to

Fig. 2. The RLC system and its optional peripherals (a voltage follower and

a diode demodulator). By design the 100 X resistor presents a negligible

inductance (2.2 lH). The inductor is made by winding and stacking a wire

(diameter Ø is 0.5 mm) uniformly around a plastic cylinder (ؼ 30 mm). Its

length is 50 mm and the number of stacks is 10. The capacitor is a common

polymer capacitor. RD¼ 2.9 MX and CD¼ 15 nF.

Fig. 1. The three configurations and associated challenges.

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DW ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi@W

@XDX

� �2

þ @W

@YDY

� �2

þ :::

s: (11)

Note that the quality factor QC is not equal to the quality fac-tor Q associated with the transfer function in which OUT isthe resistor voltage. This voltage is equivalent to the massvelocity for the mechanical oscillator. However, the differ-ence decreases with decreasing b and remains under 0.5%for b less than 0.05.

1. Option 1: Harmonic excitation

The compliance or susceptibility given by Eq. (8) is meas-ured by imposing a harmonic regime. With this condition,INT(t)¼ e0sin(xatþ/IN) is an eigenfunction of the differen-tial operator associated with the LTIS. Consequently, theoutput signal frequency and shape are identical to the inputsignal frequency and shape. Thus since the output isOUTT(t)¼ s0sin(xatþ/OUT), one can measure the ampli-tude ratio or gain (s0/e0¼ jH(x¼xa)j) and the phase(h¼Arg[H(x¼xa)]¼/OUT �/IN) by measuring the delay dof OUTT relative to INT signal (h¼dxa).

After measuring the gain and phase over several frequen-cies, the experimental Bode diagram can be plotted and com-pared with the expected theoretical curves G1 and h1 displayedin Fig. 3. This experimental procedure is well-known, yet fas-tidious and often limited to the gain characterization. In thiscase, students often use the following shortcut: after settingthe generator amplitude using the peak-to-peak value App

D dis-played on its front panel, they connect this device to the RLCinput port, visualize only the OUTT(t) signal on the oscillo-scope and finally compute the “gain” by dividing the outputamplitude by AD ¼ App

D =2.

By considering a frequency sweep, we obtain the doubleadvantage of quickly illustrating interesting experimentalbehaviour while avoiding students’ boredom and impatience.With a frequency sweep, the electromotive force generatoreemf(t) is Eemf sin½xðtÞt� with xðtÞ ¼ xðtþ TSWÞ a periodicsawtooth function whose pattern is

ðxSTOP � xSTARTÞt

TSWþ xSTART; (12)

between [0,TSW].When time increases from 0 to TSW, the circular frequency

x varies linearly from xSTART to xSTOP causing a variationof the output sine wave amplitude according to the frequencygain dependence.

The first experiment is performed with TSW¼ 9.9 ms. Bytrial and error, a suitable frequency domain [5 kHz, 20 kHz]including a resonant frequency fr* is found (see upper curveon Fig. 4). The observed discontinuities (points A and B) orig-inate from new pattern generation: at these points, the fre-quency switches “instantaneously” from xSTOP to xSTART

causing an amplitude jump since jH(xSTOP)j and jH(xSTART)jare different. These discontinuities are used to convert thehorizontal time scale to a frequency scale. Using the verticaland horizontal cursors displayed on the oscilloscope screen,the coordinates of points A–E can be found and used to mea-sure fr* along with an effective quality factor

Q�C ¼fC

fE � fD

� ��3dB

: (13)

Important differences (see Table II) between measuredand expected values for resonant frequency, gain, and qualityfactor are observed and must be therefore explained and cor-rected. The following causes are studied:

—influence of the sweep rate,—procedure to compute the gain,—influence of the oscilloscope and wires (coaxial cables),—component modelling.

2. Frequency sweep considerations

When the positive envelope of the upper curve of Fig. 4 iscompared to the expected curve G1 displayed in Fig. 3, weobserve a slight shape distortion and a resonant frequencyhigher than the expected one (see Table II). If the direction ofthe sweep is changed (fSTART¼ 20 kHz and fSTOP¼ 5 kHz),the shape changes and cannot be deduced by “reversing” theprevious shape. Furthermore, the resonant frequency is nowless than the expected value. These effects are well-knownin modal analysis9,10 or swept spectrum analysis and design11

and can be quantitatively explained by performing the follow-ing experiment.12,13 The frequency sweep is replaced by theperiodic wave train illustrated in Fig. 5 (upper curve). Each

Table I. Estimation of the parameters. The value of each parameter is given in the second row and the estimated accuracy in the third row.

fn ¼ xn

2p fp ¼ xp

2p fr ¼ xr

2p b s ¼ 1bxn

Tp ¼ 2pxp

Hmax Q QC

12.190 kHz 12.181 kHz 12.173 kHz 0.0375 348.00 ls 82.09 ls 13.34 13.33 13.29

0.259 kHz 0.259 kHz 0.258 kHz 0.0010 11.76 ls 1.75 ls 0.53 0.35 0.35

Fig. 3. Experimental transfer function superposed to two RLC transfer func-

tions associated to two distinct values of R but with the same L and C.

Curves G1 (gain) and h1 (phase) are associated with R¼ 100 X. With

Reff¼ 140.5 X the curves G2 and h2 fit the experimental transfer function

(white and black points). Phase h is always negative, varying from 0 to

�180� as the frequency increases, and is equal to �90� for f¼ fn for all

damping values.

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period of 3.5 ms duration has two distinct parts. The first iscomposed of a sine wave of 25 cycles at 14 kHz followed by asecond part represented by a null signal. The frequency fa of14 kHz is chosen to be slightly higher than the resonance fre-quency fr. The lower curve is the solution of Eq. (5a) withINT(t)¼U(t)sin(xat)U(25Ta � t). For values of time withinthe interval [0, D¼ 25Ta] (with Ta¼ 1/fa), we obtain theresponse to INT(t)¼U(t)sin(xat) which is a superpositionof the free damped response (homogeneous solution) withthe forced response (particular solution) of the formAsin(xatþ/). Notice that D has been chosen so that only theforced response remains at the end of the sine wave train. Inthis time region, the gain and phase associated with fa can bemeasured. Immediately after time t¼D, the response returnsto equilibrium in a free decay manner. The settling times forforced response and free damped response are both functionsof s. However, the damped frequency fp of the free response isdifferent from, and independent of, the forced frequency fa.

This implies that during frequency sweep each frequencybelonging to [fSTART, fSTOP] must be maintained for a dura-tion greater than s to ensure a forced harmonic response.Since TSW is not much greater than s, this condition is notsatisfied. By increasing the sweep time TSW by a factor of200, these effects are eliminated and we obtain the result dis-played in Fig. 6 (black curve OUTa). In this case, the dis-crepancy between the resonant frequency and the expected

value computed from L, R, and C values is less than 1% (seeTable II).

3. Generator setup and impedance matching

The results mentioned in Table II show that the previousprecaution does not correct the gain mismatch (factor 2) atresonance. Thus, other sources of error must be investigated.As mentioned previously, the gain was computed by dividingthe output amplitude by AD. This procedure is questionablebecause AD is not equal to the RLC input amplitude jINF(x)jas shown by the following relations:

AD ¼ Eemf

Load

Loadþ RG; (14)

jINFðxÞj ¼����Eemf

ZeðxÞZeðxÞ þ RG

����: (15)

The amplitude AD is a constant because the generator imped-ance RG is set by the manufacturer to 50 X and the valuesEemf and Load are user-defined (the “High Z” generatoroption fixed a value of Load well-above RG). On the otherhand, our RLC has been dimensioned to exhibit a frequencydependent input impedance Ze(x) that reaches values of thesame order as RG near the resonance.

Fig. 4. Frequency sweep performed with the generator directly connected to

the RLC system. The upper and lower curves represent the RLC output and

input signals, respectively. Generator settings: “Load” ¼ High Z,

AppD ¼ 2:0 V, “Sweep” mode, fSTART¼ 5 kHz, fSTOP¼ 20 kHz, Sweep

time¼ 9.9 ms, Offset¼ 0.0 V. Oscilloscope settings: External trigger. Do

not select the average function!

Table II. System identification from the analysis of the frequency response.

Assumptions R¼ (100.0 6 0.3) XL¼ (17.4 6 0.5) mH

C¼ (9.797 6 0.01) nF

fr (Hz) Hmax QC b

Figure 3 curve G1 (Expectation) 12181 6 259 13.34 6 0.53 13.29 6 0.35 0.0375 6 0.0010

Figure 4 (Fast sweep) 13157 6 11 6.80 6 0.08 7.31 6 0.07 0.0629 6 0.0011

Figure 6 black curve OUTa (Low sweep without follower) 12133 6 21 6.98 6 0.09 6.50 6 0.01 0.0787 6 0.0017

Figure 6 grey curve OUTb (Low sweep with follower) 12133 6 21 9.40 6 0.10 8.91 6 0.12 0.0561 6 0.008

Fig. 5. Response (bottom curve) to a periodic wave train (top curve). This

oscillogram illustrates the settling of a forced sinusoidal regime (note the

weak beats during the non-resonant burst) and the relaxation towards zero

through damped oscillations. Generator settings: AppD ¼ 2:0 V,

“Load”¼High Z, Sine wave, Frequency¼ 14 kHz, “Burst” mode, Burst

period¼ 3.5 ms, Cycles¼ 25, Offset¼ 0.0 V.

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Therefore, the input signal amplitude varies significantlynear resonance as shown on lower curve in Fig. 4 or blackcurve INa in Fig. 6. Consequently, the correct gain computedwith the input amplitude as a reference will differ from thegain computed using AD (nearly equal to Eemf). In otherwords, the OUTT(t) envelope is not proportional or represen-tative of the gain because the input amplitude is not constant.This behaviour indicates an impedance matching problem.14

Detailed analysis shows that fr* is slightly less than frand that QC* is lower than QC. Therefore using (fr*, QC*)instead of (fr, QC) always results in underestimated values.This effect can be corrected by decreasing the output gener-ator impedance. In that case the new RG must be less thanZe(x) for all frequencies in order to fulfil the conditionjINFðxÞj ffi Eemf ffi AD.

The solution adopted hereafter uses an operational ampli-fier (TL071) working as a follower and inserted between thegenerator and the RLC system (see Fig. 2). With this activedevice the “High Z” generator option gives an amplitude AD

equal to the input amplitude because the follower provides aunit gain, a very high impedance input, and a very lowimpedance output. As usual, care must be taken to uselow enough signal levels and low frequencies to prevent sat-uration or a slew rate of the TL071.14 The grey curves INband OUTb in Fig. 6 illustrate the follower effect on both theinput and output signals.

With the help of a diode demodulator (see Fig. 2)connected to the RLC output, the positive envelope of theRLC output signal (now equal to gain if App

D is set to 2 V) canbe extracted (upper thin line in Fig. 6) and exploited (seeTable II). Good demodulation requires a rDCD time constantgreater than 2p/xSTOP and less than TSW.

Finally, we should mention that the information displayedin Fig. 6 is limited to the gain jH(x)j. The phase could beobtained using a very slow frequency sweep by reading thephase measurement from the oscilloscope or a lock-in ampli-fier, or deducing it from the measured gain with the help of aKramers-Kronig relation.15–17

Of course, like the gain, only a few values of phase meas-ured on the oscilloscope are necessary to provide a fast iden-tification of an RLC system. Indeed, the condition h¼�90�

is obtained when the excitation frequency matches the natu-ral frequency fn and the rate of change of the phase at fn givesthe quality factor Q according to

Q ¼ � fn

2

dhdf

� �fn

: (16)

4. Influence of oscilloscope and cables

With the previous improvements, the gain mismatch isreduced to 30% which is still too high. Since the signals areanalyzed with an oscilloscope using coaxial cables, theirinfluence needs to be studied. As the manufacturer pointsout, the oscilloscope is not an ideal voltage probe. Its inputimpedance is finite and modelled by a resistor R0¼ 1 MX inparallel with a capacitor C0 � 11 pF. Taking into accountthis model, Eq. (5a) must be replaced by

LðCþC0Þd2OUTT

dt2þ L

R0

þRCþRC0

� �dOUTT

dt

þ 1þ R

R0

� �OUTTðtÞ ¼ INTðtÞ: (17)

Furthermore, at low frequencies, the Heaviside distributed-parameters cable model is reduced to a lumped capacitanceÐ k

0Ccabledx with Ccable¼ 100 pF/m and k¼ 1 m (cable length).

Thus C0 may be replaced by C0þ kCcable. Using the numeri-cal values of L, C, R, R0, C0, and kCcable, the resonant fre-quency, gain and QC factor are reduced, respectively, by0.5%, 2.1%, and 2.0%. Thus, we conclude that the gain dis-crepancy is far from being explained by a loading effect com-ing from the oscilloscope or the cables.

5. Doubts on component modelling

The relation (8) fits the experimental Bode diagram if a re-sistance equal to 40.5 X is added to R (see curves G2 and h2

in Fig. 3). The new effective total resistance Reff (140.5 X) ischecked by measuring the resonance voltage V across the100 X inductor-less resistor R. Indeed, at the resonance fre-quency f¼ fn, the input impedance must be real and equal toReff. Thus V (proportional to the current) is in phase withIN and the total effective resistance is given by R times the(IN/V) amplitude ratio. The measurement gives Reff¼ 140.8 X( 61%). Therefore, the inductor and capacitor models mustnow be examined. An ohmmeter (operating at 0 Hz) connectedto the inductor gives a value of 9.90 X. This value is confirmedby the RLC meter because an inductor quality factor equal to10.8 is measured at fW¼ 1 kHz thus giving a resistance equalto 9.79 X. The remaining resistance of 30.5 X could thereforebe attributed to the capacitor. All capacitors exhibit some re-sistance called “equivalent series resistance” RESR. It repre-sents the sum of the ohmic losses of the dielectric, materials,and connections present in the capacitor. Its value is frequencyand capacitor-type dependent. This last effect is easilyobserved by temporarily replacing the inexpensive polymercapacitor by a more expensive mica capacitor of the samevalue. Since RESR is unusually high the polymer capacitor

Fig. 6. Frequency sweep with and without the follower. The sweep is per-

formed with the generator directly connected to the RLC system (OUTa

and INa) or indirectly via the follower (OUTb and INb). The use of the

diode demodulator plugged into the RLC output gives the RLC output signal

envelope (upper thin line associated to OUTb). Generator settings:

AppD ¼ 2:0 V, “Load”¼High Z, “Sweep” mode, fSTART¼ 5 kHz, fSTOP

¼ 20 kHz, Sweep time¼ 2 s, Offset¼ 0.0 V. Oscilloscope settings: External

trigger; average function is deactivated; use of the “Save Trace SetUp” option.

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datasheet (WIMA MKS2) is examined and additional meas-urements are performed:

—At fW¼ 1 kHz, the maximum dissipation factor (loss tan-gent) is equal to 0.008. Thus, the highest expected value forRESR ¼ tanðdÞ=ð2pfWCsÞ is 130 X (d is the phase of thecomplex impedance of the capacitor).—A high series resistance of the polymer capacitor is con-firmed by the RLC-meter. It provides a loss tangent equal to0.004 thus the serial resistance is equal to 65 X.

These results could suggest significant losses inside the ca-pacitor. However, this hypothesis is invalidated because thegain curve of the low-pass filter made of this capacitor andthe resistance R (¼ 100 X) fits a theoretical “RC” gain curvedrawn with RESR¼ 0 X. Furthermore, tests performed onsimilar RLC circuits give similar results. This underlines theneed, often met in modal analysis, to introduce an effectiveloss parameter obtained from tests. The goal is to reproducethe global behaviour of our system with a model where allsources of energy dissipation are extracted from the dipolesand combined into a single effective resistance Reff.

6. Option 2: Heaviside excitation

The step response [Eq. (6)] is obtained by considering aperiodic positive square wave excitation varying from 0 toApp

D . The frequency fa of the square wave is decreased pro-gressively from 15 kHz to 250 Hz while monitoring the out-put on the oscilloscope screen. During this procedure, theoutput shape is interpreted with the help of the Bode diagram(see Fig. 3) and the Fourier series decomposition of thesquare wave input made of the odd frequency components(2nþ 1)fa of amplitude 2App

D =½ð2nþ 1Þp�. When the excita-tion frequency is reduced to 250 Hz, the output OUTT(t)decays completely between successive input steps. Usingadequate time scale adjustments, a single step response withthe corresponding excitation are isolated and stored in an in-ternal memory of the oscilloscope for future use. Accordingto Eq. (6), the pseudo-frequency is equal to fp and the ampli-tude of the mth overshoot Am (see Fig. 7) is given by

Loge

Am � E0

E0

� �¼ �ð2m� 1Þpbffiffiffiffiffiffiffiffiffiffiffiffiffi

1� b2p : (18)

This leads to an estimate of b via linear regression fromwhich

fr ¼ fp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 2b2

pffiffiffiffiffiffiffiffiffiffiffiffiffi1� b2

p : (19)

Results are given in Table III.

7. Option 3: Dirac excitation

By switching the generator from the “Square” to the“Pulse” mode the user can decrease the 50% duty cycle ofthe previous square wave to low values (typically less than1%). The RLC system is thus excited by a periodic train ofrectangular pulses. Since rectangular pulses are not Diracpulses, fundamental questions arise about the influence ofshape, amplitude, and time width of the input pulse repre-senting the Dirac pulse. Intuitively, the pulse width is thecritical parameter. It must be lower than the characteristictimes s and Tp but not too short in order to inject enough

Fig. 7. Validation of the impulse response. This response must be equal to

the derivative of the step (Heaviside) response. The impulse candidate is

graphically validated as soon as the zeros of its associated response coincide

with the step response extrema. A pulse width of 3 ls is required. The aver-

age oscilloscope function is activated.

Table III. System identification from the analysis of the responses to nonharmonic signals.

Assumptions R replaced by Reff¼ (140.8 62) XL¼ (17.4 6 0.5) mH

C¼ (9.797 6 0.01) nF

fr (Hz) Hmax QC b

Figure 3 curve G2 (Expectation) 12156 6 258 9.48 6 0.38 9.45 6 0.25 0.0528 6 0.0014

Figure 7 (Heaviside response) 11999 6 8 9.26 6 0.06 9.25 6 0.06 0.0541 6 0.0004

Cross-correlation (Gaussian) 2Vpp 33220 A (BW¼ 9 MHz) 8.105 pts 11974 6 360 7.70 6 0.18 7.69 6 0.18 0.065 6 0.002

Figure 8 Cross-correlation (Gaussian) 33220 A and SR560 (2Vpp,

BW¼ 1 MHz and Gain¼ 50) DAQ¼ 8.105 pts 11981 6 360 9.02 6 0.13 9.01 6 0.13 0.0555 6 0.0008

Cross-correlation (Gaussian) 2Vpp 33521 A (BW¼ 100 kHz) 8.105 pts 11980 6 360 8.67 6 0.10 8.67 6 0.10 0.0570 6 0.0010

Cross-correlation (PRBS) 2Vpp 33521 A (PN11, 40 kbps) 104 pts 11980 6 300 8.68 6 0.21 8.66 6 0.10 0.0570 6 0.0010

Figure 9 (Impulse response curve 2) 12149 6 38 9.39 6 0.11 9.38 6 0.11 0.0551 6 0.0002

Figure 9 (Amplitude spectrum curve 3) 12150 6 20 9.08 6 0.38 9.07 6 0.19 0.0551 6 0.0002

Figure 10 [Impulse response (a)] 11997 6 170 9.49 6 0.01 9.48 6 0.02 0.0528 6 0.0001

Figure 10 [Amplitude spectrum (b)] 11842 6 52 9.41 6 0.70 9.39 6 0.35 0.0532 6 0.0061

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energy (proportional to the product of the width by thesquare of the amplitude). When these conditions are fulfilledthe pulse shape influence weakens. The generator built-infunctions offer rectangular, triangular, or sine pulses to eas-ily verify this.

However, a quantitative criterion must be defined to vali-date the pulse width. Validation comes from comparisonbetween the impulse response and the Heaviside response.Since the first is the derivative of the second, the maxima ofthe Heaviside response must occur at the zero crossing of theimpulse response. Students are required to reload to the os-cilloscope the previously stored step response and compare itto the tested impulse response. The example shown in Fig. 7illustrates the procedure with a valid rectangular excitationpulse.

The impulse response can be analysed according to Eq. (7)using the pseudo-frequency equal to fp and the quantity

Loge

Bm

B1

� �¼ �ð2m� 2Þpbffiffiffiffiffiffiffiffiffiffiffiffiffi

1� b2p ; (20)

with Bm the mth overshoot. This leads to estimate b vialinear regression thus fr with Eq. (19). Results are given inTable III.

8. Option 4: Noise excitations

This solution takes advantage of Eq. (3a) by using an exci-tation whose autocorrelation CIN,IN(t) is a Dirac function.Random signals (“Noise”) offer this feature, since any non-zero time shift of such a signal is perfectly uncorrelated withthe unshifted signal. Equations (3b) and (1a) therefore leadto CNoise,OUT(t)¼ hT(t).

Note the contrast between the Dirac and the noise excita-tions. The first requires a short duration and high amplitudesusceptible to cause a nonlinear behavior, whereas the sec-ond can be applied with lower amplitudes over a longerduration.

In preparation for this experiment students must first readthe DAQ datasheet to know its main specifications thenselect two input channels on the terminal block and finallyconfigure the LABVIEW program by setting parameters appear-ing in the front-panel.

The experimentation begins by applying a Gaussian whitenoise whose spectrum ranges from 0 to 9 MHz as mentionedin the Ag33220A user’s guide. Next this signal and the corre-sponding output are sampled, cross-correlated, and the resultis compared to hT(t).

The expected shape and characteristic frequency areobtained if the user performs and averages many acquisitionsover an acquisition time significantly greater than s andremoves the DC components of both input and output digi-tized signals prior to cross-correlation.

However, the damping factor b is overestimated. The dis-crepancy is reduced by increasing moderately both the gen-erator amplitude and the number of averages. A significantimprovement could be obtained by concentrating the energyin a reduced bandwidth. Since this is not possible with theAg33220A, we add a low noise voltage preamplifier (SR560Stanford Research) acting also as a configurable filter. It hasbeen set to reduce the original noise bandwidth from [0, 9MHz] to [0, 30 kHz], the frequency domain where H(x) is“non-zero,” while increasing the excitation level in this newbandwidth. The Ag33521A combines features of theAg33220A generator and SR560 preamplifier and offers anew type of noise known as a pseudo-random bit sequence

Fig. 8. Identification using Gaussian noise or PRBS noise as input. (a) Noise input shapes in the time domain. (b) Noise autocorrelation. (c) Cross-correlation

between the noise input and the corresponding output. (d) Zoom on cross-correlation.

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(PRBS). Good results with the PRBS option are obtained(see Table III) if the bit generation rate is high compared to1/s, many bits are generated during the characteristic time s,and the total time needed to generate the whole sequence ishigher than the acquisition time. All these conditions are ful-filled if one chooses, for instance, the PN11 option (thepseudo-random sequence contains 211� 1 bits), a generationrate of 40 kbits/s, a sampling frequency of 200 kHz, and only10000 points to acquire (see Table III).

Typical curves are displayed in Fig. 8. Curve (a) displaysboth the white noise and the PRBS input signals which arevisually quite different. However, they produce similar auto-correlation functions and flat amplitude spectra in the do-main of interest. Curve (b) displays a typical input signalautocorrelation, whose Dirac-like shape validates the randomcharacter of the noise. Curve (c) displays the correlation ofthe output with the input, and curve (d) gives a zoom on thecentral region.

B. Validation of the LTIS fundamental relations

In the previous experiments, the fundamental excitations(harmonic, Heaviside, Dirac, noise) were applied to the RLCquadrupole considered as a second order system. Analysis ofthe responses or computed signals in the time domain leadsto the RLC “black box” identification through the numericalestimates of b and xr. However, experiments validatingdirectly the fundamental relations such as the relationbetween hT(t) and H(x) or Eq. (1b) or (2) are still missing.They are presented hereafter.

1. Computing the transfer function

If a single Dirac excitation is applied to the RLC circuit,then, according to theory, the Fourier transform hF(x) of theresponse hT(t) is the transfer function H(x). To check thisassertion and at the same time compare two spectrum analy-sers, the output is simultaneously digitized and FFT-transformed by both the oscilloscope and the DAQ using thesame parameters (fs, N). Both analyzers are triggered by the

Sync generator signal, the oscilloscope is configured to the“Single shot” mode, and the generator is set to the “Pulse”and “Burst” modes. Analysis of the graphs displayed inFig. 9 (oscilloscope) and Fig. 10 (DAQ) allows testing thestudents’ knowledge of basic signal processing.18 More spe-cifically they need to understand the full details enclosed inthe relation N¼ fsTacq, where N is the number of sampledpoints, fs¼xs/2p is the sampling frequency, and Tacq is theacquisition time. This implies a mastery of the samplingeffect associated with fs and the finite acquisition time effectrelated to N. Moreover, it draws the students’ attention to thenature and setting of an anti-aliasing filter but also to the ex-citation characteristics.

2. Sampling effect

To examine the sampling effect, we consider an infiniteacquisition time where the original signal hT(t) is replacedby the sampled signal h�TðtÞ ¼ hTðtÞ

Pþ1�1 dðt� nTsÞ whose

Fourier transform is

h�FðxÞ ¼ hFðxÞ� fs

Xþ1�1

dðx� nxsÞ ¼ fsXþ1�1

hFðx� nxsÞ:

(21)

Fig. 10. Sampled impulse response and its FFT using a DAQ. The first 200

samples over 1000 are displayed in the time domain. Horizontal scaling of

the digitized signal (a) and its spectrum (b) is obtained by multiplying,

respectively, the sample number by the time resolution (Ts¼ 1/fs) and the

frequency resolution (fs/N). The spectrum shows the folding symmetry at

sample number 500 associated with fs/2. DAQ settings: fs¼ 200 kHz,

N¼ 1000, Input limits¼60.5 V, Input configuration¼ “single referenced,”

Window (FFT analysis performed with the rectangle window). Generator

settings: see Fig. 9.

Fig. 9. Impulse excitation (a) and the corresponding response displayed in

the time domain (b) or the frequency domain (c). Generator settings:

“Pulse” and “Burst” modes activated, Manual trigger, “Load”¼High Z,

Pulse width¼ 3 ls, AppD ¼ 2:0 V, Offset¼ 1.0 V. Oscilloscope settings: gen-

eral (external trigger, single shot mode, acquire mode¼ “high resolution”),

FFT (fs¼ 200 kHz, Span¼ 10 kHz, Center¼ 12.2 kHz, Scale¼ 2 dB/div,

Offset¼�42.6 dBV, Window¼Rectangle).

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The real-valued signal hT(t) produces a symmetric ampli-tude spectrum jhF(x)j relative to 0 Hz. Therefore, the spec-trum of the sampled signal h�TðtÞ, obtained by periodizing(period equal to xs) the spectrum of the original signal andadding all the patterns, will display a folding or mirror fre-quency at multiple of xs/2 known as the Nyquist circular fre-quency. The folding frequency is clearly visible in Fig. 10(b)displaying all the raw Fourier data given by LABVIEW. The linkbetween the sample number and the frequency is given by

Frequency ¼ Sample number � ðfs=NÞ: (22)

Thus data are displayed in the [0, fs] interval in contrastwith the [0, fs/2] interval selected by the oscilloscope’s man-ufacturer. The distance between hFðxÞ ¼ HðxÞ and its aliasobviously decreases with decreasing fs, leading to an overlapas soon as fs/2 becomes less than the upper frequency fmax

contained in hT(t). The overlap is called aliasing and leads toan effective folding of all frequencies greater than theNyquist frequency into the Nyquist domain [0, fs/2]. Figure 3suggests a value of 30 kHz for fmax and Fig. 10(b) shows thataliasing is not present here, thus validating our choice for fs.Aliasing may in some cases produce a hatch signal but thisvisual criterion is often unreliable. This can be demonstratedby sampling over many periods a 100 Hz sine wave with asampling rate close to the signal frequency (for example,fs¼ 101 Hz). In that case a perfect but fictitious 1 Hz(¼�100þ 101) sine wave will appear. Signals presentingfast rising time and thus high fmax are sources of troublebecause fs is instrument limited (typically 2 GHz for our os-cilloscope and 1 MHz for the DAQ). Furthermore, even if werespect the Nyquist criteria, fs/2 greater than fmax, problemsare expected because noise frequencies greater than fs/2 canalways be folded back to the Nyquist domain. To limit thisproblem, an antialiasing device must be inserted between thesignal and the sampler/holder of the analyser. This low-passanalog filter would not affect the frequencies lower than fs/2and eliminate those higher than fs/2. Since such a low passfilter cannot be built (causality violation), one has to use areal filter presenting a gradual roll-off in the transition band.Filter choice (Butterworth, Tchebychev, etc.) depends on thedesired response characteristics. Whatever the choice, theuser must select a cut-off frequency fc using a criterion. Thisfrequency must be greater than fmax to preserve the originalsignal and less than fs/2 to avoid aliasing. Note that the am-plitude attenuation is already 30% at fc if the �3 dB criterionis adopted. Of course if excessive noise components are pres-ent in the transition band, aliasing cannot be avoided.

The peaks appearing in Fig. 3 (curve G2), Fig. 9 (curve c),and Fig. 10(b) show similar characteristics (see Tables II andIII) plotted with different vertical scales: logarithmic (dB)for the oscilloscope (see Fig. 9) and linear for the others. Asusual, care must be taken to use a sufficiently high frequencyresolution (small xs/N value) compared to the peak width inorder to represent it correctly. This directly affects the qual-ity of measured values of fr, QC, or b, An improvement isobtained by fitting the peak with the help of an interpolationLorentzian function based on Eq. (8).

3. Windowing effects

As the acquisition time is finite, only a portion of the sig-nal is analysed. Mathematically, the original hT(t) is thusreplaced by a windowed sampling signal h�TðtÞPðtÞ, where

the function PðtÞ is null everywhere except in the acquisitioninterval [0, Tacq] where it is equal to unity.

In the frequency domain, this rectangular window leads tothe convolution of the h�FðxÞ spectrum with the sinc functionTacqsinðxTacq=2Þ=½xTacq=2�. This produces energy leakageconcentrated in the lobes coming from the sinc function.

In the time domain, the original signal must now be con-sidered periodic with the pattern hTðtÞPðtÞ. Thus if the origi-nal signal does not go to zero at the limits of the windowPðtÞ, the periodic signal will be discontinuous.

To reduce this effect, the rectangular window is replacedby another function. In the time domain, the goal is to mini-mize the discontinuities while in the frequency domainobtaining more rapidly attenuated lobes than the sinc func-tion. Numerous windows exist, each suited for a particularuse. All of those windows come with the drawback that theamplitude of the original spectrum is modified. In this study,the windows offered by the oscilloscope (Rectangle, Han-ning, Flat Top, Blackman-Harris) have been tested and com-pared. The amplitude spectrum associated with therectangular window gives the best agreement concerning thequality factor because the analysed signal dies away beforethe limits of the selected acquisition time interval.

4. Excitability considerations

Relation (2) cannot be blindly used to deduce HðxÞ fromthe ratio OUTFðxÞ=INFðxÞ because the excitation spectrummust be non-zero at all frequencies where the transfer func-tion is to be deduced. This condition is verified by a Diracexcitation, whose spectrum is constant over all frequencies.If this condition is not fulfilled then the ratio will not produceH(x). This problem is well-known in modal testing andexplains important changes observed in the measured ratioOUTFðxÞ=INFðxÞ or sound emitted by a tuning fork whendifferent hammer tips are used.

5. Computing the output

In the previous experiment, the digitized impulse responseis stored in an ASCII file (IMP.dat). Convolution of the digi-tized impulse response with a given digitized excitationshould give the output signal according to Eq. (1a). To checkthis, students are first asked to inject one period only of a3 kHz sine wave excitation. Both input and output signals arevisualized on the oscilloscope. The oscillogram named “thereference screen” is shown in Fig. 11. Then students are askedto digitize the input signal with the DAQ controlled by a LAB-

VIEW application. The input digitized signal is stored in a sepa-rate ASCII file (IN.dat) then convolved with the first fileIMP.dat. The result is displayed on the computer screen bythe LABVIEW application and compared to the output signal dis-played on the oscilloscope (lower curve in Fig. 11). If the twofiles IMP.dat and IN.dat are created using suitable and identi-cal acquisition parameters, a close agreement is obtainedbetween the calculated curve and the expected curve (lowercurve in Fig. 11). This validates relation (1 b), and Fig. 11shows that the signal delivered by any sensor is a modifiedimage of the original signal applied to the sensor. More spe-cifically we observe a time stretching and a shape distortion.

6. Deducing the excitation from the response

The challenge here is to recover the input signal from theoutput and the impulse responses. This is a recurrent problem

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in instrumentation where the user knows only the output andthe sensor properties embedded into the impulse response.The solution comes from deconvolution, a key area in signaland image processing. In contrast with the previous para-graph, an indirect method called the Fourier-quotient methodis used and numerically implemented with the followingformula:

fINTðkTsÞg0;::;N�1 ¼ FFT�1 FFT½OUT:dat�FFT½IMP:dat�

� �: (23)

The LABVIEW program is organized in three sequences:

— The first, common to all programs used in this lab ses-sion, digitizes a signal (here the output), displays thesampled values on a graph [see Fig. 12(a)], and stores thedata in a text file (here OUT.dat).

— The second loads the digitized impulse response from theIMP.dat file and displays it [see Fig. 12(b)].

— The third computes the input signal with the above for-mula and displays it [see Fig. 12(c)].

The computed input signal is clearly the expected one (aperiod of a truncated sine wave). It can be a bit noisy, butagreement with the expected curve can be significantlyimproved by increasing the sample rate, averaging manyacquisitions, and filtering or smoothing19 the average. Ofcourse each change in the acquisition parameters (fs, N) usedto acquire the OUT.dat file requires updating the IMP.dat fileto ensure re-sampling the impulse response with the sameparameters. It may be tempting to replace FFT[IMP.dat] bysampled values obtained directly from the transfer functiongiven by Eq. (8) and therefore eliminate the need of oneFFT. However, this is incorrect because as explained previ-ously the transfer function does not take into account any ofthe sampling effects (aliasing, leakage).

IV. CONCLUSIONS AND CLASSROOM

IMPLEMENTATION

A milestone lab session has been described. It uses stand-ard instruments and the familiar damped harmonic RLC os-cillator as a test bench for teaching the basics of

instrumentation and signal processing. This one-degree-of-freedom (1-DOF) system is best suited to discover throughexperiments the generic properties of model systems knownas LTIS. The study of such systems involves three quantities:the input signal, the output signal, and a system specificfunction that can be also expressed both in the time and fre-quency domains. Depending on the available information,the user must identify the system, predict the output signal,or recover the input signal. Answers have been given byusing and validating various excitations (standard or moreelaborate) and careful numerical implementation of convolu-tion, correlation, and Fourier transform. The suggestedexperiments lead students to analyse datasheets, improvetheir knowledge of instrument settings, and deal with the in-evitable discrepancies between experimental and expected

Fig. 11. Response to a transient. The transient excitation describing a single

period of a 3 kHz sine wave (upper curve) is applied to the RLC system. The

lower curve represents the associated response. This oscillogram is called

the “reference screen.” Generator settings: “Pulse” and “Burst” modes acti-

vated (Manual trigger). “Load”¼High Z, AppD ¼ 3:0 V, Offset¼ 0 V.

Fig. 12. Experimental deconvolution using the Fourier-quotient method.

The output (a) and the impulse response (b) signals are digitized

(fs¼ 200 kHz, N¼ 1000) then used to recover the excitation. The period of

the reconstructed signal (c) is marked with a vertical dashed line located

between sample numbers 66 and 67. Using Eq. (21), we find a frequency

between 2.99 and 3.03 kHz, in agreement with the expected result (see cap-

tion of Fig. 11).

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results. This prepares them to question their models or assessthe influence of instruments (impedance matching) or proto-cols or settings, especially those associated with FFT.

The analogy between the RLC oscillator and the mechani-cal mass-spring damped system has been highlighted andused to illustrate common pitfalls met during modal analysis.As modal analysis is based on the hypothesis that a N-DOFlinear system can be decomposed in N 1-DOF linear sys-tems,6 the fundamental interest of the system studied here isjustified.

To cover all the material presented in this paper, theteacher must split it into various lab sessions. The harmoniccharacterisation of the RLC system including the carefulimplementation of the frequency sweep is well-adapted tofirst and second year students (Figs. 3–6). Alternativeapproaches to system identification and validation of theLTIS fundamental relations can be performed in 4 h and arehighly appreciated by more advanced students (Figs.7–12).20 For engineering-oriented students, an implementa-tion of digital filtering to enhance the signal to noise ratiocould be performed.19 Furthermore, the careful examinationof the sources of errors could be an opportunity to elaboratea course on error analysis.2

ACKNOWLEDGMENTS

The authors gratefully acknowledge Patrick Moretto-Capelle, Gilles Bailly, Christophe Buet, and Vincent Boitierfor fruitful discussions.

APPENDIX: GRAPHICAL ANALYSIS PROCEDURE

The main objective is to compare results obtained withdifferent methods (test of reproducibility). The first outputsof a global analysis of uncertainties are the time, the fre-quency, and the amplitude resolutions. As the accuracy ofmeasurements can be altered by many factors (see text), theestimation of these resolutions is complex and time consum-ing. Therefore, a simple and fast graphical analysis of uncer-tainties is performed. For example, an impulse-type responsedisplayed on the computer screen (Figs. 8 and 10) is ana-lyzed by clicking on the first ten extrema. The softwarerecords their coordinates and computes (hTpi, DTp) fromthe abscissa and (hbi, Db) from the ordinates. From thesevalues, the averages and the standard deviations associatedto fr, QC, and Hmax are deduced and compiled in the tables.

The quantity Hmax is deduced from Figs. 4 and 6 by dividingthe maximum peak-to-peak output value by the peak-to-peakamplitude generator display App

D when the “High Z” option isselected.

1“GUM, guide to the expression of uncertainty in measurement,” <http://

www.bipm.org/en/publications/guides/gum.html>.2H. J. C. Berendsen, A Student’s Guide to Data and Error Analysis (Cam-

bridge U.P., 2011).3A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill,

New York, 1962).4R. Bracewell, The Fourier Transform and its Applications, 3rd ed.

(McGraw-Hill, New York, 1999).5P. M. Morse and K. U. Ingard, Theoretical Acoustics (Princeton U.P.,

Princeton, NJ, 1987).6A. Girard and N. Roy, Structural Dynamics in Industry (Wiley, New York,

2007).7P. E. Dupuis, “Essais de vibrations, Mesures et exploitation des resultats,”

Techniques de l’Ingenieur, traite Genie mecanique, BM 5160 (2000).8J. R. MacDonald and M. K. Brachman, “Linear-system integral transform

relations,” Rev. Mod. Phys. 28(4), 393–422 (1956).9C. Lalanne, Mechanical Vibration and Shock Analysis, 2nd ed., Sinusoidal

Vibration (Wiley, New York, 2009), Vol. 1.10J. A. Lollock, “The effect of swept sinusoidal excitation on the response of a

single-degree-of-freedom oscillator,” presented at the 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 2002.

11Agilent Technologies, Spectrum Analysis Basics: Application Note 150(Agilent Technologies, Santa Clara, CA, 2006), <http://cp.literature.agilent.

com/litweb/pdf/5952-0292.pdf>.12V. Leroy, M. Devaud, and J. C. Bacri, “The air bubble: Experiments on an

unusual harmonic oscillator,” Am. J. Phys. 70(10), 1012–1019 (2002).13P. Sandoz, E. Carry, J. M. Friedt, B. Trolard, and J. G. Reyes, “Frequency

domain characterization of the vibrations of a tuning fork by vision and

digital image processing,” Am. J. Phys. 77(1), 20–26 (2009).14P. Horowitz and W. Hill, The Art of Electronics, 2nd ed. (McGraw-Hill,

New York, 1989).15H. Bode, Network Analysis and Feedback Amplifier Design (D.Van Nos-

trand Company, Inc., New York, 1945), Chap. 14.16F. W. King, “Numerical evaluation of truncated Kramers-Kronig trans-

forms,” J. Opt. Soc. Am. B 24(7), 1589–1595 (2007).17J. L. Passmore, B. C. Collings, and P. J. Collings, “Autocorrelation of elec-

trical noise: An undergraduate experiment,” Am. J. Phys. 63(7), 592–595

(1995).18Agilent Technologies, The Fundamentals of Signal Analysis: Application

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agilent.com/litweb/pdf/5952-8898E.pdf>.19J. S. Bendat and A. G. Piersol, Random Data: Analysis and Measurement

Procedures (Wiley, New York, 1971).20Performed by third year students in “Physique fondamentale” and

“Physique & Applications” at the Paul Sabatier University. Substantial

preparation and post session homework is expected. Previous laboratory

and course sessions provide all the necessary background in mathematical

concepts (including the FFT) and instrumentation.

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