ET 304b
Laboratory 5
SuperpositionTheorem With Ac and Dc Sources
Objective: Use the Superposition
theoremto find the total responseto sinusoidalac and dc
inputs. Use an oscilloscopeto measureac waveswith a dc offset. Measurecompositesignals
with digital multimetersand determinethe rms valuesof the ac and dc components.
Theoretical Background
The superpositiontheoremis a techniquefor solving electric circuits that have more than
one source. Nodal and meshanalysisalso can find solutionsfor complex networksthat include
multiple sources,but requirethe solutionof an entire set of voltagesor currents. Another
limitationof thesemethodsis that the sourcesmust all be of the samecurrenttype: ac or dc. The
superposition
theoremdoesnot havethis limitation,which makesit a valuabletechniquein
electroniccircuit design.
The superposition
theoremonly appliesto linearcircuitsand linearcircuit responses.
A
circuit must containlinearelementsto be linear. A linearcircuit elementhasa proportional
outputfor a given input. Componentsthat follow Ohm'slaw are all linear elements.All
componentscoveredin the lecturequalify as linearelements,includingdependentsourcesthat
havea linearrelationshipbetweentheir outputandthe controllingparameter.Linear responses
arethe currentthroughany branchand the voltagedrop acrossany component.Powercan not
be found directlyby superposition
becauseit is a non-linearfunctionof currentor voltageand
the componentvalue.
Superposition
is implemented
by first identifyingthe circuitresponse
of interest.This is
the voltageor currentfor which we will solvethe circuit. Unlike the other networktechniquesit
will not be necessary
to solvefor othervariablesto find this response.The procedurefor using
superposition
is:
l.) de-activeall independentsourcesexceptone. Replacede-activatedvoltagesources
by shortcircuitsand de-activated
currentsourcesby opencircuits.
2.) Calculatethe desiredresponseof the circuit for the remainingsource.
3.) RepeatstepsI and2 until responses
are found for all sourcesin the circuit.
4.) Add the individualresponsefrom eachsourcesto get the total responseto all the
circuit sources.
Superposition
is especiallyusefulin finding the responseof a circuit to combinedac and dc
inputs. This is typicalin electroniccircuits.
Electroniccircuitsuseactivecomponentssuchas transistorsand integratedcircuitsto
amplify ac signals. A dc voltageis necessary
for the activecomponentsto operate.The output
is generallya compositesignalthat hasboth a dc and ac component.[n practiceit is not possible
to de-activatethe dc sourceand measurethe ac responsebecausethe dc bias is necessaryfor
circuit operation.It is possibleto measurethe dc outputonly with the ac input de-activated.
The oscilloscopecan effectivelymeasurecompositeac and dc signals. It is also possible
to measurethe ac componentseparately
usingthe oscilloscope.To isolatethe ac signalfrom the
dc level,the verticalcouplingswitch mustbe in the properposition.The verticalcouplingswitch
is locatedbelow the verticalinputsand hasthreepositions.ac, gnd, dc. The gnd position
groundsthe input. The scopechannelis referencedto zerowith the input in this position. The
S p r i n g2 0 0 2
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other two positionsdeterminehow the input signalcouplesto the scopesvertical input amplifier.
Figure I showsthat with the switch in the ac positionthe input connectsto the vertical amplifier
through a capacitor,C6.
input
Vert.Amp.
To vertial
deflection
circuits
FigureL VerticalCouplingSwitchCircuit.
The capacitorblocksthe dc componentof the input signalwhile passingthe ac signal. Ac
couplingmakesmeasurement
of low level ac voltagespossibleeven if a significantdc
componentis present.
Moving the switchto the ac positionwill block the dc componentand centerthe ac wave
aboutthe groundpositionon the display. The ac portioncan now be amplifiedif necessaryby
decreasing
the V/div scalingof the scopechannel.Placing
the couplingswitch in the dc position
displaysboth the dc levelsand the superimposed
ac level.
Ac couplingwill centerany waveformthat hasa dc component,evenpulses. The pulse
in Figure2-a goesfrom 0 to +V when dc coupledto the scope.Ac couplingcentersthis wave
aboutthe groundreference,as shownin Figure2-b This is not the correctdisplayof the wave
sinceit only goespositive.
a. )
b.)
Figure 2. Effects of Ac ScopeCoupling on positivc Voltage pulses.
Digital multimeterscan alsomeasurecompositesignals,but meterconstructionand the
type of waveformare a consideration.True rms digital multimetersdisplaythe rms values,or
effectivevalue,of the total ac and dc componentsof the signal. The rms uulu" of time vaiying
wavesdependson the shapeof the signal. True rms metersaccuratelyrespondto any waveform
and displaythe correctrms value. Averagerespondingmultimeterdesignsassumesinusoidalac
inputsand calibratethe outputto readcorrectly.Measuringa non-sinusoidal
ac wave with an
averagerespondingmultimeterproduceserroneousresults.
Before measuringac waveswith a DMM, determinewhetherthe instrumentis a true rms
or averagerespondingdevice. Also determinethe frequencyresponselimit of the instrument.
Instrumentinstructionmanualsusuallycontainthis information.
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Procedure
Before starting the lab, install 10x probes to both channelsof the scopeand
compensatethem. Obtain frequency responseinformation about the DMM used for
making the following measurements.Also determine if the DMM is true rms or average
responding.
1.)
Constructthe circuit in Figure3. Use the sourcetransformationtheoremto find an
equivalentac voltage sourcefor the given currentsource.
V,r 12Vdc
C
lrr
0.02uF
2.5mApeak
f=5.0kHz
Figurc 3. Lab Circuit l.
2.)
3.)
4 .)
s.)
6.)
De-activatethe 12 Vdc sourceand replaceit with a shortcircuit accordingto the
superposition
theorem. Measurethe resultingvoltageacrossthe capacitor,V" usingthe
scopewith dc couplingand recordthe resultin Table 1. Make the samemeasurement
usinga DMM and recordthe resultin Table 1 Find the rms value of the scope
measurement
by usingthe formulatV,n,,:0.707Vp"ur.
De-activatethe equivalentac voltagesourceand replaceit with a short circuit. Activate
the dc sourceand measurethe dc voltageappearingacrossthe capacitorwith a
oscilloscopeand DMM. Keep the scopein the dc couplingmode. Recordthese
measurements
in Table l.
Activateboth voltagesourcesin the circuit. With the scopein the dc couplingmode,
measurethe voltageacrossthe capacitor.Sketchthe traceon the grids provided.
Determinethe dc level by measuringthe verticaloffsetfrom ground.
Switchthe scopeinput to ac couplingand observethe waveformof V". Sketchthe trace
on the grids provided. Measurethe peakvalueof the wave and convertit to rms by using
the formula in step2.
Solvethe circuit in Figure! usingthe superposition
theorem.Find the rms value of the ac
responseand the dc level. Enterthesevaluesin Table l. Calculatethepercentageerror
betweenthe theoreticalvaluesand the measuredvaluesof the scopeand the DMM
measurements
usingthe following formula:
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Discussion
Points
Whatis the procedurefor findingvoltagesandcurrentsusingsuperposition?
How doesthe
verticalcouplingeffectthe displayof the measured
voltage?Whattypeof couplingshouldbe
dc signalswith the scope?Low levelacwith highleveldc? Ac voltagesonly?
usedto measure
Do thetheoreticalvaluesmatchthe measured
valueswithin component
tolerances?Whattypeof
DMM is usedfor the experiment:
responding?
truermsor average
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Table l. Circuit I Measurements
for Circuit 1'
Table2. Enor Percentages
o/oErrorDMM
o/oErrorScoPq
Vc (ac rms)
Vc (dc)
Circuit2 Rc:
Table 3. Circuit 2 Measurements
Enor Circuit2'
Table4. Percentage
o/nF.rror DMM
o/oErrorScoPe
Vn (ac rms)
Vn (dc)
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