ac circuits lecture 20 - ac...ac circuits maximum currents & voltages phasors: a simple tool...
TRANSCRIPT
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ACCircuitsMaximumcurrents&voltages
Phasors:ASimpleTool
Electricity & MagnetismLecture 20
Today’sConcept:
Electricity&Magne?smLecture20,Slide1
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Othervideos:
Prof.W.Lewin,MIT—LookonYouTube
MechanicalUniverse,ACCircuits
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Practical Test Schedule
Newschedule
• Firstsession:Friday12:30to13:20
•AthroughHou• SecondsessionMonday12:30to13:20
•HuangthroughPeralta• ThirdsessionMonday14:30to15:20
•PukanichthroughZ• D200sec?onwilldoitFriday
•AthroughTrujilloat15:30.•VajahaththroughWat16:30
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DC Measurements
Voltage:! Ideallyavoltmeterhasinfiniteresistance,inrealityitsresistanceisabout10MΩ.
! Connectthemeterinparallelwiththecomponent! MakesureDCVscaleischosen(startfromhighestscaleandworkdown.)
! Makesuretheposi?velead(red)ispluggedintotheVinput
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DC Measurements
Current• Ideallyanammeterhaszeroresistance,inrealityitsvariesfrom0.1Ωto1kΩfromthe10Ascaletothe400µAscale
• Connectthemeterin'series'withthecomponent
• CheckthatDCAscaleischosen
• Theposi?ve(red)leadneedstobepluggedintoacurrentinput
• Startusingthe10Ainputandscalethenchangetoalowercurrentscaleifneeded
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DC Measurements
Resistance! Componentmustbeisolatedfromcircuittogetavalidresistancereading
! avoidtouchingthemeterprobeswhilemeasuringtheresistance:yourbodyresistancecanbe100kΩormore.
! Disconnec?ngoneendofcomponentisusuallyadequate! UsethesameinputasforVmeasurement
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AC Measurements
UsingtheOscilloscope! CH1andCH2groundsarecommonandmustbeconnectedtobeconsistentwithgroundofthefunc?ongenerator
! Triggeringcontrolstabilizesthedisplay.Assuresthatthesignalisatthesamevoltageand'slope'athetriggerposi?ononthescreen,indicatedbyaT.
! UseDCcouplingunlessoneneedstodisplayasmallsignalontopofaDCoffset.
! RemarkthatwhencomparingDMMvoltagemeasurementstothescope,theDMMreadsRMSvoltagewhichisthe
• amplitude/√2or• Vpp/2√2
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Uncertainties
DCMeters! Digitalerrorofatleast±1digitintherightmostdigit.! Analog'calibra?on'errorisspecifiedbyapercentageofthereading,oienabout1%
! Thesetwosourcesoferroradd(notinquadrature).! Forexampleiftheerrorisspecifiedas5%±1digitthenareadingof1.02Vwouldhaveerrorof±(.05+.01)=±.06V
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Uncertainties
Oscilloscope! DigitalmeasurementssubjecttobothdigitalandanalogerrorsasforDMM(Ifunknownuse±(3%+1digit).
! Formeasurementswithcursors,movethecursorsandnotetheminimumincrementofthereading
! Ifyoureadthescalebyeye,usethesametypees?matesasyouusewhenreadingaruler:oien±½smallestdivisionbutusediscre?onforverylargeorsmalldivisions,andalsotakeintoaccountthelinewidthandpossiblenoiseonthesignal.
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Circuit Technique
58 CHAPTER 6. INTRODUCTORY ELECTRONICS NOTES: PRACTICE
Figure 6.1: Bad and Good breadboarding technique.
• Try to build your circuit so that it looks like its circuit diagram:
– Let signal flow in from the left, exit on the right (in this case, the “signal” is justV ; the “output” is just I, read on the ammeter);
– Place ground on a horizontal breadboard bus strip below your circuit. When youreach circuits that include negative supply, place that on a bus strip below theground bus.
– Use colour coding to help you follow your own wiring: use black for ground, redfor the positive supply. Such colour coding helps a little now, a lot later, whenyou begin to lay out more complicated digital circuits.
Figure 6.2 shows bad and good examples of breadboard layouts. Figure 6.3 showsthe layout of a typical breadboard. Typically, one places components in the middlegroups with vertical interconnects and power lines and grounds in the horizontalinterconnects at top and bottom.
Figure 6.2: Bad and good breadboard layouts of a simple circuit
Bad
Goodugly!
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Bad
ugly!
Good and Bad component layout
58 CHAPTER 6. INTRODUCTORY ELECTRONICS NOTES: PRACTICE
Figure 6.1: Bad and Good breadboarding technique.
• Try to build your circuit so that it looks like its circuit diagram:
– Let signal flow in from the left, exit on the right (in this case, the “signal” is justV ; the “output” is just I, read on the ammeter);
– Place ground on a horizontal breadboard bus strip below your circuit. When youreach circuits that include negative supply, place that on a bus strip below theground bus.
– Use colour coding to help you follow your own wiring: use black for ground, redfor the positive supply. Such colour coding helps a little now, a lot later, whenyou begin to lay out more complicated digital circuits.
Figure 6.2 shows bad and good examples of breadboard layouts. Figure 6.3 showsthe layout of a typical breadboard. Typically, one places components in the middlegroups with vertical interconnects and power lines and grounds in the horizontalinterconnects at top and bottom.
Figure 6.2: Bad and good breadboard layouts of a simple circuitConnections among pins in the breadboard.
Use horizontal rows for voltage busses: +5V, ±12V, gnd.
Use vertical rows for connecting components
together.
Good
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`
58 CHAPTER 6. INTRODUCTORY ELECTRONICS NOTES: PRACTICE
Figure 6.1: Bad and Good breadboarding technique.
• Try to build your circuit so that it looks like its circuit diagram:
– Let signal flow in from the left, exit on the right (in this case, the “signal” is justV ; the “output” is just I, read on the ammeter);
– Place ground on a horizontal breadboard bus strip below your circuit. When youreach circuits that include negative supply, place that on a bus strip below theground bus.
– Use colour coding to help you follow your own wiring: use black for ground, redfor the positive supply. Such colour coding helps a little now, a lot later, whenyou begin to lay out more complicated digital circuits.
Figure 6.2 shows bad and good examples of breadboard layouts. Figure 6.3 showsthe layout of a typical breadboard. Typically, one places components in the middlegroups with vertical interconnects and power lines and grounds in the horizontalinterconnects at top and bottom.
Figure 6.2: Bad and good breadboard layouts of a simple circuit
+5V bus
gnd bus
to +5V ofpower supply
to gnd ofpower supply
to scope
connection
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Resistors
R I = VR/R = Vmax/R sin(ωt)
Amplitude=Vmax/R
ε = Vmaxsin(ωt)
Electricity&Magne?smLecture20,Slide3
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Capacitors
C I = VmaxωC cos(ωt)
whereXC=1/ωCislikethe“resistance”ofthecapacitorXCdependsonω
Amplitude=Vmax/XC
I = dQ/dt
Q = CV = CVmaxsin(ωt)
90o
ε = Vmaxsin(ωt)
Electricity&Magne?smLecture20,Slide4
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Inductors
L I = − (Vmax/ωL) cos(ωt)
Amplitude = Vmax/XL
whereXL=ωLislikethe“resistance”oftheinductorXLdependsonω
dI/dt = VL = Vmaxsin(ωt)
90o
ε = Vmaxsin(ωt)
Electricity&Magne?smLecture20,Slide5
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RL Clicker Question
AnRLcircuitisdrivenbyanAC generatorasshowninthefigure.
ForwhatdrivingfrequencyωofthegeneratorwillthecurrentthroughtheresistorbelargestA)ωlargeB)CurrentthroughRdoesn’tdependonωC)ωsmall
L
R
XL = ωL
Asω → 0, sodoesXL
As ω → 0, resistanceofcircuit→Rcurrentgetsbigger
Electricity&Magne?smLecture20,Slide6
impedance
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R Imax = Vmax/R VR inphasewithI
L Imax = Vmax/XL VL 90o aheadofIXL = ωL
C Imax = Vmax/XC VC 90o behind IXC = 1/ωC Currentcomesfirstsinceit
chargescapacitor
Becauseresistorsaresimple
Oppositeofcapacitor
Likeawireathighω
Likeawireatlowω
Summary
Electricity&Magne?smLecture20,Slide7
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Vmax = Imax XL
Vmax = Imax XC
Vmax = Imax RV inphasewithI
V 90o behind I
V 90o aheadofI
MakessensetowriteeverythingintermsofIsincethisisthesameeverywhereinaone-loopcircuit: Imax XL
Imax XC
Imax R
Phasorsmakethissimpletosee
Alwayslooksthesame.Onlythelengthswillchange
εmax
“Doyouhaveanyfancy-schmancysimula?onsfortoshowme?”
Electricity&Magne?smLecture20,Slide8
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AC Circuit Simulations
hvp://www2.epsd.us/robo?cs/phet/en/simula?on/circuit-construc?on-kit-ac.html
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Imax XL
Imax XC
Imax R Imax XL
Imax R
Imax XC
εmax
Butnowweareaddingvectors:
Imax XL
Imax XC
Imax R
εmax
The Voltages still Add Up
Electricity&Magne?smLecture20,Slide9
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Imax XL
Imax XC
Imax R
εmax
Make this Simpler
Imax XL
Imax XC
Imax R
Electricity&Magne?smLecture20,Slide10
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Imax R
εmax = Imax ZImax(XL − XC)
Make this Simpler
Imax XL
Imax XC
Imax R
Electricity&Magne?smLecture20,Slide11
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Imax R
εmax = Imax Z
Imax(XL − XC)
Make this Simpler
Electricity&Magne?smLecture20,Slide12
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Imax R
εmax = Imax Z
Imax(XL − XC)
R(X
L − XC )
φ
φ
ImpedanceTriangle
Make this Simpler
Electricity&Magne?smLecture20,Slide13
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φ
R (XL − X
C )
VCmax = Imax XC
VLmax = Imax XL
VRmax = Imax R
Summary
Imax = εmax / Z
εmax = Imax Z
Electricity&Magne?smLecture20,Slide14
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Imax XL
Imax R
εmax
Example: RL Circuit Xc = 0
Electricity&Magne?smLecture20,Slide15
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ABC
DrawVoltagePhasors
CheckPoint 2
Imax XL
Imax R
εmax
Electricity&Magne?smLecture20,Slide16
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0
15
30
45
60
1
ABC
CheckPoint 4
DrawVoltagePhasorsImax XL
Imax R
εmax
Electricity&Magne?smLecture20,Slide17
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0
13
25
38
50
1
TheCURRENTisTHECURRENT
ABCD
φ isthephasebetweengeneratorandcurrent
φ
CheckPoint 6
Imax XL
Imax R
εmax
Electricity&Magne?smLecture20,Slide18
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CheckPoint 8
0
13
25
38
50
1
ABC
Whatdoesthevoltagephasordiagramlooklikewhenthecurrentisamaximum?
IXc
IRε
IXL
IXc
IR
εIXL
Electricity&Magne?smLecture20,Slide19
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0
13
25
38
50
1
Whatdoesthevoltagephasordiagramlooklikewhenthecapacitorisfullycharged?
ABC
IXc
IRε
IXL
IXc
IR
εIXL
CheckPoint 10
Electricity&Magne?smLecture20,Slide20
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CheckPoint 12
Whatdoesthevoltagephasordiagramlooklikewhenthevoltageacrosscapacitorisatitsposi?vemaximum?
ABC
IXc
IRε
IXL
IXc
IR
εIXL
Electricity&Magne?smLecture20,Slide21
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ConceptualAnalysisThemaximumvoltageforeachcomponentisrelatedtoitsreactanceandtothe
maximumcurrent.Theimpedancetriangledeterminestherela?onshipbetweenthemaximum
voltagesforthecomponentsStrategicAnalysis
UseVmaxandImaxtodetermineZUseimpedancetriangletodetermineRUseVCmaxandimpedancetriangletodetermineXL
C
RLV
Calculation
Electricity&Magne?smLecture20,Slide22
ConsidertheharmonicallydrivenseriesLCRcircuitshown.Vmax = 100 VImax = 2 mAVCmax = 113 VThecurrentleadsgeneratorvoltageby45o
LandRareunknown.
WhatisXL,thereactanceoftheinductor,atthisfrequency?
~
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CompareXLandXCatthisfrequency:
A)XL < XC B)XL = XC C)XL > XC D)Notenoughinforma?on
Thisinforma?onisdeterminedfromthephaseCurrentleadsvoltage
VL = ImaxXLVC = ImaxXC
VR (phaseofcurrent)
VL
VC V leads
IR
V
45ο
Calculation
C
RLV
Electricity&Magne?smLecture20,Slide23
ConsidertheharmonicallydrivenseriesLCRcircuitshown.Vmax = 100 VImax = 2 mAVCmax = 113 VThecurrentleadsgeneratorvoltageby45o
LandRareunknown.
WhatisXL,thereactanceoftheinductor,atthisfrequency?
~
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WhatisZ,thetotalimpedanceofthecircuit?
A)B)C)D)35.4 kΩ50 kΩ 21.1 kΩ70.7 kΩ
Calculation
C
RLV
Electricity&Magne?smLecture20,Slide24
ConsidertheharmonicallydrivenseriesLCRcircuitshown.Vmax = 100 VImax = 2 mAVCmax = 113 VThecurrentleadsgeneratorvoltageby45o
LandRareunknown.
WhatisXL,thereactanceoftheinductor,atthisfrequency?
~
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WhatisR?A)B)C)D)
Z = 50 kΩ
sin(45°) =0.707
cos(45°) =0.70735.4 kΩ50 kΩ 21.1 kΩ70.7 kΩ
= 50 kΩ × 0.707
= 35.4 kΩ
Calculation
C
RLV
DeterminedfromimpedancetriangleR
Z = 50kΩ
(XC − XL)45ο R = Z cos(45o)
Electricity&Magne?smLecture20,Slide25
ConsidertheharmonicallydrivenseriesLCRcircuitshown.Vmax = 100 VImax = 2 mAVCmax = 113 VThecurrentleadsgeneratorvoltageby45o
LandRareunknown.
WhatisXL,thereactanceoftheinductor,atthisfrequency?
~
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A)B)C)D)
Z = 50 kΩ
R = 35.4kΩ
WhatisXC?
35.4 kΩ50 kΩ 21.1 kΩ70.7 kΩ
VCmax = ImaxXC
Calculation
ConsidertheharmonicallydrivenseriesLCRcircuitshown.Vmax = 100 VImax = 2 mAVCmax = 113 VThecurrentleadsgeneratorvoltageby45o
LandRareunknown.
WhatisXL,thereactanceoftheinductor,atthisfrequency?
~
C
RLV
XL = XC − R
XL = 56.5 kΩ − 35.4 kΩ
R
Z(XC − XL)
45ο
Westartwiththeimpedancetriangle:
Electricity&Magne?smLecture20,Slide26
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Practical Test HintsYouwillhave
! 4bananaplugwires,~4alligatorclips! 1ScopeProbe(x1/x10)! 1BNCwire! adaptors:Tee,BNC-MaleBanana,BNC-FemaleBanana! proto-board(Useit!),somesmallwires! scope,func?ongenerator,DMM,specsheet,mini-grapplers! 4baveriesandholderorDCpowersupply
• Firstsession:Friday12:30to13:20
• AthroughHou
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