Transientcurrentsandvoltages
BasicACcircuitelements
InaDCcircuit,theelectro‐motiveforcespushtheelectronsalongthecircuitand
resistorsremovethatenergybyconversiontoheat.InACcircuits,currentsvaryin
time,sowehavetoconsidervariationsintheenergystoredinelectricandmagnetic
fieldsofcapacitorsandinductors,respectively.
Capacitance:thecapacitanceCbetweentwooppositechargedsurfacesisdefined
by:
Q V

(1)
C
Where:Qisthemagnitudeofthechargedistributedoneithersurface,andVisthe
potentialdifferencebetweenthesurfaces.
DifferentiatingEq.(1)andusingI=dQ/dt,weobtain:
dV
I (2)

dt C
Inductance:Theusualmodelforaninductorisacoil(solenoid).ByFaraday’sLawof
self‐inductance,achangingcurrentinacoilinducesabackelectro‐motiveforce
(emf)thatopposesthechangeincurrent:
dI
V L
(3)
dt
WhereVisthebackemfacrosstheinductor,dI/dtisthederivativeofthecurrent
throughtheinductorandListheinductance.
TheRCCircuit
InanRCcircuitwithnosignalsource,weassumethecapacitortobeinitially
charged.Thevoltageacrossthechargedcapacitoris:
Q
V0  0 (4)
C
V
(5)
Thisvoltageproducesaninitialloopcurrent I 0  0 R
dV V
ByusingKirchhoff’sloopruleandthederivativeofOhm’slaw: C
  0 ,the
dt R
voltageacrosstheresistor(andcapacitor)isgivenby:
 t / RC
 V0 e  t / V (t )  V0 e
(6)
1
Theconstantτ=RCiscalledthetimeconstantofthiscircuit.Itdefinesthetime
neededforV(t)tofallto1/eofitsinitialvalue.
TheRLCcircuitwithonlythecapacitorandtheinductorshowsaresonant
behaviorasthecapacitordischarges,currentisvariableandtheinductorresponds
bycreatinganinducedemf.
Whenaresistorisaddedtothecircuit,itdissipatedenergy,sothattheoscillation
becomesdamped.
Figure 1: LC circuit and
oscillation of voltage in time.
TheExercise
Youwilluseacircuitboardwithseveralresistors,capacitorsandinductors(BoxA,
below):
Figure2:BOXA
ADataAcquisitionSystem(DAQ)willprovidetheACvoltagesneededinyour
experiment.Atthesametime,theDAQwillcollectdatafromthecircuit(Voltage,
Time)andsendthemtothecomputer.
SomeoftheDAQportsarealreadyconnected.Donotchangethem.
2
YouwillusetheAnalogOutput(ao0)portastheACvoltagesourceforyourcircuit.
ConnecttheBNCwireatthetopoftheT‐connectoronao0.
YouwillalsocollectthedatafromthecircuitthroughtheAnalogInput(ai0)port.
DatainrealtimewillbedisplayedthroughaLabViewapplication.
TheRCcircuit
a i0
Figure3:TheRCCircuit
diagram
ao 0
BuildaRCcircuitaccordingtoFigure1,usingC=1.0μFandtheRvar(Potentiometer
5‐25kΩ).OpentheDesktopfolder:2ndYrLabFilesandselectChargeandDischarge
ofaCapacitor(LabViewApplication).
TheAnalogOutputworksasafunctiongenerator.UsetheSquareWaveformand
varythefrequencyuntilyouobserve2‐3cyclesofcharging/dischargingofthe
capacitor.Inthebackgroundofthecharge/discharge,youmaynoticethesquare
wave(lightgrey).
Thewholedatasetwillhaveabout10000datapoints,whichisalittletoomuchfor
yourPythonprogram.TrimmingthedatasetispossiblewithExportData,by
confirmingthelengthediting.Cursorsarenowavailable.Placethematthetwoends
ofonedecay(orrise)andclickonExportDataagain.Savethetextfileonyour
memorystick.
ThedatasethasTime(s)inthefirstcolumnandAmplitude(Volt)inthesecond.
Datawasacquiredaswithanoscilloscope:zeroamplitudeisatthemiddleofthe
graph.Youhavetoadjust(lift)alltheamplitudevaluessothatallarepositive:find
thelargestnegativeamplitudeandadditsabsolutevaluetoallthedata.
WriteafittingprogramwithstatisticaloutputtofittheRCtransientvoltage.Output
thetimeconstantτandcalculateR(Rvar).Estimateuncertainties.
ComparethecalculatedvaluewithadirectmeasurementofRorRvarusinga
multimeter.Note:disconnectR(Rvar)fromthecircuitbeforeconnectingthe
instrument!
TheRLCcircuit
ModifytheRCcircuitfromFigure1byinsertinganinductorinserieswithRandC.
UseRvar(Potentiometer5‐25kΩ),C=1.0μFandL=1.0mH.
3
UsetheSquareWaveformontheAnalogOutput:chooseafrequencyaround200Hz.
ConnecttheAnalogInputtotheinductoronly.Observetheresonanceoscillation
betweenthecapacitorandtheinductor,dampedbythepresenceoftheresistor.
Dataisacquiredaswithanoscilloscope,butforthisexerciseyouareinterestedin
findingtheperiodofthedampedoscillation,soyoudon’thavetolifttheamplitude
values.
Trimthedatasettoabout300datapoints(onedampedoscillation)anddetermine
theperiodT(usethefirst2‐3periodsofoscillation).Comparewiththeresonant
angularfrequencyωr:
2
1
r 

T
LC
WrittenbyRuxandraSerbanescu(2012‐2013).TheDataAcquisitionSystemwasbuiltbyLarry
Avramidis.LarryalsowrotetheLabViewapplication.
References:L.R.Fortney–PrinciplesofElectronics:AnalogandDigital,Harcourt
BraceJovanovich1987
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