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/. - Spring2002 (1) f.j+l exp404b.doc 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 Spring2002 exp404b.doc 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 Spring2002 exp404b.doc 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. Spring2002 exp404b.doc + + + **+ + rh:l#il+ iH I i't, WovC Sowri Fiq{r ff ff ++ IJ L 1,, Y YI!' ++a+ lalrov Pvtct I ge ,/Div t+ I lr "f- rl * 1rolDir +I il l pelOtv 50 rH! SiG r{ov! I pry'0iv _v_\ _t50 rHr (ol Conpqqord '-r'- r TTflN dntl l I m:/Oiv ++i +i !; .] .\ -Yl -i 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: Spring2002 exp404b.doc 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. Spring2002 exp404b.doc 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. Spring2002 exp404b.doc 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) Spring2002 exp404b.doc 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? Spring2002 exp404b.doc Table I High SourceResistanceMeasurements.lx probe h SourceResistance Measurements: lOx probe firi(est.) C 0.02uF 0 . 0 1uF 0.005uF Spring2002 Table3 Effectsof Shunt u C MeasuredValues t, (pS) firi l0 tanceF5.0 .U kH KITZ TheoreticalValues (pS) t, Gr; exp404b.doc ittl Channel1 Volts/div ttl ltl ChannelI Volts/div Spring2002 ttll tttl ttl ttl T ttl L ttl tt ttl rl - -tttl Jlll Channel2 Volts/div 11 ttl Time/div Channel2Volts/div tttl ttl tttl Jll ttl l tl ttl ttl l tl ttl tll ttl Time/div exp404b.doc tttl tttl trl ltl tl tl tttl ttf ttL -ttl tl l Jtl tttl tttl tttl tttl ttl tttl l ttl __ Channel1 Volts/div tttl tttl Channel I Volts/div Spring2002 Channel2Volts/div ttl ttl ttl tttl ttl ttt tttt rrrL Time/div l l tl ttl ttl tttl tttl Jlll ttl tl tl tttl tttl Channel2Volts/div t2 Time/div exp404b.doc