ET 304b
Laboratory 4
RC Circuits: FrequencyResponseand Rise time
Objective: Observetransientresponseeffectsfor RC circuits excitedwith squarewave inputs.
RelateRC time constantsto the sinusoidalbandwidthof a RC circuit. Compensateoscilloscope
probesto achievevoltage division independentof frequency. Apply the RC circuit conceptsof
rise time, frequencyresponse,and sourceimpedanceto the oscilloscopemeasurementproblem.
Theoretical Background
The RC combinationsin amplifiers and other electroniccircuits control the high frequency
responseof the system. Complex networkscan be reducedto simpleRC circuits for analysisif a
dominatetime constantexists.The dominatetime constantis the slowestresponseto abrupt
changesin the circuit. It hasa valuemuchgreaterthan all other circuitRC combinations.
Rg,
The simplifiedmodelbelow showsa signalgeneratorwith an input resist&flce,
parallel
Ri
combinationof amplifier input resistance
connectedto an idealamplifierthroughthe
C1.
and capacitance,
Figure l. Simplificd Timc RcsponseModcl of an Amplificr
The voltagedivider formedby the componentsRg,Ri, and Ci presentsa fractionof the generator
voltageto the amplifierinput. The capacitorin parallelwith the input resistancemakesthis
voltagedivider dependenton frequency.The input voltagein termsof the circuit impedancesis:
I n, t t x ^ )
Y = v ^l * R, x . ,
ll
' [ R,
-l J
givesthe amplifierinput in termsof the
Substitutingin the definitionof capacitivereactance
resistance
divider
and capacitance.
\r/. -
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(1)
f.j+l
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Equation 1 showsthat a resistivedivider determinesthe input voltage. The third term of the
product is a lowpassfilter characteristicwith the cutofffrequency determinedby equation2.
f" = ---7-]------1
Hz
2 n . (R, ll R, / . C,
(2)
The cutofffrequencyfiri, is the frequencywhereV, is reducedto a level of 0.707times the
maximum value determinedby the resistivedivider network. The frequency,fs;, governsthe
high frequencyresponseof the overallcircuit.
EquationsL and2 definethe Figure 1 circuit responseto sinusoidalinputs. If V, is a
squarewave or a squarepulse,Equation3 determinesthe value of Vi. The input and output
voltasesare now functionsof time.
( n
)f
Y(t) = v"(t)l:-:=
-')
(3)
lll-""' |
" " (R, * R,
/[
)
The circuittime constant,
r11iis givenby t,,, = (*, ll R* ). C, This is the time constantof the
input circuit that controlsthe overallcircuit responseto rapidlychanginginputs.
Sincethe voltageacrossthe input capacitance,
C;, c&nnot changeinstantaneously,
the
input voltagewill not follow the abrupttransitionsof pulseinputs. After sometime, Ci will
chargeto a final value. If the periodof the pulsewave is too short,then Vi will neverreachits
final value. If trri is much shorterthanthe periodof the pulse,the input to the circuit will reach
its final value. The resistivedivider of equation4 determinesthe final value.
,.(
V , = V ^ l-l
R
)
(4)
I R, + R, ]
The rise time of the pulsewave appliedto Figure I relatesto the sinusoidalcutoff
frequencythroughthe relationship:
0.35
t- - _
'
f*,,
(s)
t.: the risetime in seconds
frri : the high frequencycutoff.
This is accuratewhen the circuit hasa dominatecutofffrequency.Rise time is the time it takes
for a pulseinput to changefrom 10% to 90Yoof its final value. Figure2 showsthe rise time
for a singlepulseinput. The risetime is definedas tr:tro-tgo.This measurement
is
measurement
madeusingan oscilloscope.
Assumingthat the low frequencylimit of the circuit is dc (0 Hz), firi determinesthe
bandwidthof the circuit. Equation5 showsthat the risetime-bandwidthproductis constant.
Circuitswith fast risetimeswill havewide bandwidthsand high upper cutofffrequencies.This
also impliesthat the circuit hassmallervaluesof Ci sinceV; quickly reachesits final value.
where
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10
o
t
6
ir
a
o
o
80
Ylll' 60
40
20
0 .1
0.2
o.7
0.4
Time (sec)
Figure 2. Definition of Risc time for Pulse Wavefonns.
Conversely,Equation5 relatestime responseof the circuit to frequencyresponse.The bandwidth
of a circuit can be determinedby injectinga squarewave signalinto it and measuringthe rise
time. Computingthe fH;using equation5 shouldgive the samecutofffrequencyas applyingsine
wavesand producinga frequencyresponseplot.
Oscilloscopeinputsare examplesof how RC networkseffectthe time and frequency
responseof a circuit. The input capacitance
of the scopeandthe probecableeffectthe high
frequencyresponseof the instrument.A equivalentresistance,
R., paralleledwith a capacitance,
C,, modelsverticalinput of scopeamplifiers.The input voltageV; developsacrossthese
components.An signalsource,V* ,with an sourceresistance
,Rg,cohh€ctsthe circuit undertest
to the scopeinput.
vi
verticalamp.
:
Figure3. CircuitModclof Scopewith lx ProbeandSignalSource.
Figure 3 models the scope connectedto a signal source through a lx probe. This circuit is
similar to the RC circuit in Figure I The input capacitance of the scope increases due to the
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of
shunt capacitanceof the cable.Using a lx probe significantlyreducesthe frequencyresponse
the instrumentdue to this increasein input capacitance.
The bandwidth limit of the scopewith a 1x probe is the high frequencylimit definedby
equation2 above. In terms of the scopeparametersit becomes:
f-
^Hi
-
(6)
2 n. ( R, llR") . C"
Sine inputs abovethis frequencywill be attenuated.Pulseinputs will have a significant rise time
that can be computedfrom equation5. Equation6 showsthat the high frequencycutoffof the
instrumentdependson the signal sourcesresistance,R , and the scopesinput parameters.Low
impedancessourceswill exhibit higher cutofffrequenciesfor constantscopeparameters.It may
be possibleto accuratelymeasurelow resistancesourcesat moderatelyhigh frequencieswith a
lx probe, but measuringhigh resistancesourcesat high frequencywill introduce significant
error.
The effectsof input capacitance
and instrumentloadingcan be reducedby using an
voltage
attenuatorprobe(10x probe). A lOx attenuatorprobeis a frequencycompensated
dividerthat increasethe scope'sinput impedanceto 10 MO and almosteliminatesthe
circuit model.
capacitance
of the cableand scopeinput. Figure4 givesthe scope/probe
R"
verticalamp.
R.=9Mf)
Figure 4. Circuit Model of Scopewith 10x AttenuatorProbe.
SettingR":9 MO reducesVi by a factor of ten. Adjusting C" suchthat R.C.:R"C" removesthe
capacitiveeffects of the probe cableand the scope,which flattensthe circuit frequencyresponse
by adjustingCuwith a square
This is calledcompensating
the probe. The probeis compensated
when the dc portionof the squarewave is horizontal.
wave input. The probeis compensated
over
Figure 5 showstestwaveformsfor l0x probesthat are correctlycompensated,
compensated,
and undercompensated.
occurswhen Cuis too large. The squarewave test signalhas
Overcompensation
excessiveovershoot,which reflectsan amplificationof high frequencies.This is shown in the
sinetraceof Figure 5-b. A high frequencysinewave will displaylargerthat normalwith an
probe.
overcompensated
occurswhen Cuis too small.The test signalwill havea rounded
Undercompensation
leadingedge,which indicateshigh frequencyattenuation.High frequencysinewaveswill be
probe.This is shown in the sine
smallerthan normalwhen measuredwith an undercompensated
traceof Figure 5-c.
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rh:l#il+
iH
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ff ff
++
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1,, Y
YI!'
++a+
lalrov
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lr
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il
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50 rHe
{bl Ovacoatrr6otld
lc) Undc{cllllp.ffiofr
Figure 5. Effects of Over and Under Compensationof AttenuatorProbe.
10x probefor maximum
All scopemeasurements
shouldbe madeusing a properlycompensated
accuracy.
The effectthat the scopeprobehason the circuit dependson the impedanceof the test
circuit and the type of probeused. For low impedancesourcessuchas the signalgeneratorsin
the circuit analysismodel.
the lab, Figure6 represents
SignalGenerator
a""".'""""""""
verticalamp.
Figure 6. lx Probe Model connectedto a Low Impedance Signal Generator.
The parallelcombinationof R, and Rs is approximatelyRr. The value of C. reflectsthe probe
cablecapacitance.The value of firi is:
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frHi
-
-
:l.274lvfrIz
zn(so
lltvo[zsopn)
that eivesa risetime of :
t.=
0 .35
0.35 --275
nS
f*
T.274NftIz
Thesecalculationsindicatethat the 1x probe will give reasonablyaccuratemeasurementsup to
1.274MHz. This is much lower than the frequencylimit of the scopesin the lab.
Adding the attenuatorprobewill increasethis limit by increasingthe scope'sinput
resistanceand lower the effectivevalue of C.. Figure 7 showsthe circuit analysismodel of the
10x probeconnectedto a low impedancesignalgenerator.
SignalGenerator
R"=9Mo
verticalamp.
Figurc 7. AttcnuatorProbcand Low ImpcdanceSignal Source.
The computing fi.riand t,.gives:
=I274lvlTlz
f-
zn(so
llroMoX2.sopF)
0 35
0.35
= . 2 7 5n S
f,,,
1274 MHz
probereducesthe effectsofthe capacitance
The compensated
by a factorof 100 and increases
the bandwidthof the scopeto over I GHz In practicethis bandwidthwill be lower due to the
responseof other circuit time constants.
tr=
Procedure
Part l-OscilloscopeAttenuator Probe Compensation
l.)
2.)
Selecttwo attenuator(l0x) probesand connectedthem to the scope.
Attach channel I to the I kHz squarewave test signallocatedon the scopeface. Locate
the compensation
capacitor.It shouldbe on the probebody or on the BNC connectorat
capacitorwith a small screwdriver
the scopeend of the probe. Adjust the compensation
until a flat topped squarewave results. Repeatthis for channel2.
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3.)
4 .)
s)
6)
Using the compensatedl0x probes,setthe output of the function generatorfor a2Y
peak ac sinewave at 500 kHz.
With the signal from (3) displayed,adjustthe compensationcapacitoruntil the voltage
increases.Removethe function generatorand re-attachthe probeto the calibration
source. Sketchthe resultingwaveform and commenton its shapein the report
discussion.
Reconnectthe 500 kHz sine sourceand adjustthe compensationscrewuntil the displayed
voltagedecreases.Reconnectthe probeto the calibrationsquarewave and sketchthe
resultingwaveform. Discussthe shapein the report.
Recompensatethe probe for flat responseon the squarewave.
Part 2-Effectsof SourceResistance
7.)
8.)
Changethe function generatoroutput to a2 V peak squarewave at 500 kHz.
Add externalsourceresistanceto the 50 ohm function generatoroutput as shown in the
schematicof Figure 8. Measurethe risetime of the functiongeneratorsignalwith the
scopeprobein the lx positionfor the valuesof R, given in Table 1.
Signal Generator
verticalamp.
F500kHz
Y :2Y P
Figure 8 lx Probe Conncctcdto Function Gcneratorwith lncreasedSourceResistance.
e.)
Computethe valueof firi from the risetime measurement
and enterit in the column
labeledf11;m€osured.
Thesevalueswill includethe actualcapacitance
valuesof the probe
and scopeinput. Computethe valuesof firi and t, usingequations5 and 6 and the
estimatedvaluesC. from figure 8. Enterthesevaluesin the columnslabeledfiti est.and t,
est. From equation6 and the valuesof fi1;measured,computethe valuesof Cu"64.This
is a betterestimateof the total capacitance
in the circuit. Enter thesevaluesin the Table
also.
Repeatthe procedurein step8 with the scopein the 10x position. Figure9 showsthe
in Table2. The total source
circuit modelusedfor this analysis.Enterthe measurements
resistancenow will be the sum of R*, R', and the externalRr.
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SignalGenerator
i---""-:
R"=9Mc)
i:
verticalamp.
Figure9. High SourccRcsistance
with lOx Probe.
Part 3 - Effects of Shunt Capacitance
10.)
Constructthe circuit shownin Figure 10. Setthe functiongeneratorfor a squarewave
outputwith an amplitudeof 2 V peakat a frequencyof 5.0 kHz.
R1
1k
5.0 k H z
Figurc10. Circuitfor Part3.
View the outputvoltagewith an attenuatorscope probeand measurethe rise time of the
voltagefor eachof the capacitorvaluesgiven in Table 3.
1 2 ) Using the measuredrise time valuein Table3, computef111
using equation5 for all
capacitors.Enterthe resultsin the tableunderthe measuredfni heading.
1 3 . ) Computethe theoreticalvaluesof firi andt, usingequations6 and 5 for all C valuesis
Table3. Enter thesevaluesunderthe theoreticalheadinssof the table.
14.) Computethe percentageerror betweenthe theoreticalvaluesand the measuredvalues of
t. and f1y;.Placethesepercentagesin the lab report and commenton the agreement
betweenthe values.
11)
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I)iscussionPoints
What effect doeschangingsourceresistancehave on fui and t ? What effect doesan
increasein C have on the bandwidth and rise time of the circuits presented?What impact does
using a 10x probe have on the scopesinput resistanceand input capacitance?What effect does
using a 10x probe have on bandwidthof the combinedprobe and scopenetwork? What is the
procedurefor compensatingattenuatorscopeprobes?
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Table I High SourceResistanceMeasurements.lx probe
h SourceResistance
Measurements:
lOx probe
firi(est.)
C
0.02uF
0 . 0 1uF
0.005uF
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Table3 Effectsof Shunt
u C
MeasuredValues
t, (pS)
firi
l0
tanceF5.0
.U kH
KITZ
TheoreticalValues
(pS)
t,
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