ACCircuitsMaximumcurrents&voltages

Phasors:ASimpleTool

Electricity & MagnetismLecture 20

Today’sConcept:

Electricity&Magne?smLecture20,Slide1

Othervideos:

Prof.W.Lewin,MIT—LookonYouTube

MechanicalUniverse,ACCircuits

Practical Test Schedule

Newschedule

• Firstsession:Friday12:30to13:20

•AthroughHou• SecondsessionMonday12:30to13:20

•HuangthroughPeralta• ThirdsessionMonday14:30to15:20

•PukanichthroughZ• D200sec?onwilldoitFriday

•AthroughTrujilloat15:30.•VajahaththroughWat16:30

DC Measurements

Voltage:! Ideallyavoltmeterhasinfiniteresistance,inrealityitsresistanceisabout10MΩ.

! Connectthemeterinparallelwiththecomponent! MakesureDCVscaleischosen(startfromhighestscaleandworkdown.)

! Makesuretheposi?velead(red)ispluggedintotheVinput

DC Measurements

Current• Ideallyanammeterhaszeroresistance,inrealityitsvariesfrom0.1Ωto1kΩfromthe10Ascaletothe400µAscale

• Connectthemeterin'series'withthecomponent

• CheckthatDCAscaleischosen

• Theposi?ve(red)leadneedstobepluggedintoacurrentinput

• Startusingthe10Ainputandscalethenchangetoalowercurrentscaleifneeded

DC Measurements

Resistance! Componentmustbeisolatedfromcircuittogetavalidresistancereading

! avoidtouchingthemeterprobeswhilemeasuringtheresistance:yourbodyresistancecanbe100kΩormore.

! Disconnec?ngoneendofcomponentisusuallyadequate! UsethesameinputasforVmeasurement

AC Measurements

UsingtheOscilloscope! CH1andCH2groundsarecommonandmustbeconnectedtobeconsistentwithgroundofthefunc?ongenerator

! Triggeringcontrolstabilizesthedisplay.Assuresthatthesignalisatthesamevoltageand'slope'athetriggerposi?ononthescreen,indicatedbyaT.

! UseDCcouplingunlessoneneedstodisplayasmallsignalontopofaDCoffset.

! RemarkthatwhencomparingDMMvoltagemeasurementstothescope,theDMMreadsRMSvoltagewhichisthe

• amplitude/√2or• Vpp/2√2

Uncertainties

DCMeters! Digitalerrorofatleast±1digitintherightmostdigit.! Analog'calibra?on'errorisspecifiedbyapercentageofthereading,oienabout1%

! Thesetwosourcesoferroradd(notinquadrature).! Forexampleiftheerrorisspecifiedas5%±1digitthenareadingof1.02Vwouldhaveerrorof±(.05+.01)=±.06V

Uncertainties

Oscilloscope! DigitalmeasurementssubjecttobothdigitalandanalogerrorsasforDMM(Ifunknownuse±(3%+1digit).

! Formeasurementswithcursors,movethecursorsandnotetheminimumincrementofthereading

! Ifyoureadthescalebyeye,usethesametypees?matesasyouusewhenreadingaruler:oien±½smallestdivisionbutusediscre?onforverylargeorsmalldivisions,andalsotakeintoaccountthelinewidthandpossiblenoiseonthesignal.

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!

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

`

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

Resistors

R I = VR/R = Vmax/R sin(ωt)

Amplitude=Vmax/R

ε = Vmaxsin(ωt)

Electricity&Magne?smLecture20,Slide3

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

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

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

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

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

AC Circuit Simulations

hvp://www2.epsd.us/robo?cs/phet/en/simula?on/circuit-construc?on-kit-ac.html

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

Imax XL

Imax XC

Imax R

εmax

Make this Simpler

Imax XL

Imax XC

Imax R

Electricity&Magne?smLecture20,Slide10

Imax R

εmax = Imax ZImax(XL − XC)

Make this Simpler

Imax XL

Imax XC

Imax R

Electricity&Magne?smLecture20,Slide11

Imax R

εmax = Imax Z

Imax(XL − XC)

Make this Simpler

Electricity&Magne?smLecture20,Slide12

Imax R

εmax = Imax Z

Imax(XL − XC)

R(X

L − XC )

φ

φ

ImpedanceTriangle

Make this Simpler

Electricity&Magne?smLecture20,Slide13

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

Imax XL

Imax R

εmax

Example: RL Circuit Xc = 0

Electricity&Magne?smLecture20,Slide15

ABC

DrawVoltagePhasors

CheckPoint 2

Imax XL

Imax R

εmax

Electricity&Magne?smLecture20,Slide16

0

15

30

45

60

1

ABC

CheckPoint 4

DrawVoltagePhasorsImax XL

Imax R

εmax

Electricity&Magne?smLecture20,Slide17

0

13

25

38

50

1

TheCURRENTisTHECURRENT

ABCD

φ isthephasebetweengeneratorandcurrent

φ

CheckPoint 6

Imax XL

Imax R

εmax

Electricity&Magne?smLecture20,Slide18

CheckPoint 8

0

13

25

38

50

1

ABC

Whatdoesthevoltagephasordiagramlooklikewhenthecurrentisamaximum?

IXc

IRε

IXL

IXc

IR

εIXL

Electricity&Magne?smLecture20,Slide19

0

13

25

38

50

1

Whatdoesthevoltagephasordiagramlooklikewhenthecapacitorisfullycharged?

ABC

IXc

IRε

IXL

IXc

IR

εIXL

CheckPoint 10

Electricity&Magne?smLecture20,Slide20

CheckPoint 12

Whatdoesthevoltagephasordiagramlooklikewhenthevoltageacrosscapacitorisatitsposi?vemaximum?

ABC

IXc

IRε

IXL

IXc

IR

εIXL

Electricity&Magne?smLecture20,Slide21

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?

~

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?

~

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?

~

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?

~

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

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

• SecondsessionMonday12:30to13:20

• HuangthroughPeralta

• ThirdsessionMonday14:30to15:20

• PukanichthroughZ

• D200sec?onwilldoitFridayat

• AthroughTrujilloat15:30.

• VajahaththroughWat16:30