CopyofCircuitAnalysisStudyGuide_Cheat-sheet.docx

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Delta(Δ)-Wye(Y) Transformations:

Chapter 1

Charge=Coulombs(integralofcurrent,areaunderthecurrentcurve)

Current=Coulombs/Second(derivativeofCharge,slopeofcharge,changein

chargeovertime,dq/dt)

Voltage=Joules/Coulomb

Power:P=IV,P=(V)^2/R,P=R(I)^2

Amp-hours=Current*time

Watt-hours (Energy)=Power*time

Watt-hours * $$$=costduetoelectricity

Chapter 2

Resistance=ρ(l/A)p=resistivityconstant,l=length,A=crosssectionalarea

Ohm's Law:V=IR,I=V/R,R=V/I

Short Circuit=0ohms(Resistance),anycurrentcanflowthroughashort

Open Circuit =∞ohms(Resistance),nocurrentcanflowthroughanopencircuit

Branches,nodes,loops:B=L+N-1

Series:TwoElementsshareasinglenode.SameCurrentforallresistorsinseries,differentVoltages(Voltage

Division).

Parallel:TwoElementsconnectedtosametwonodes.SameVoltageforallresistorsinparallel,different

Currents(CurrentDivision).

Linearity:V/I=Rcreatesaperfectlylinearrelationshipforcertaincircuits.

Kirchhoff's Current Law (KCL):Currentin=Currentout

Kirchhoff's Voltage Law (KVL): ThesumofVoltagesaroundaloop=0

Voltage Division (Series Resistors):

TwoResistorsareconsideredinseriesiftheyhave

thesamecurrentpassthroughthem.IfwehaveaVs

andtworesistorsinseries,hereistheequations:

V1=i(R1)

V2=i(R2)

-V+V1+V2=0

-V+i(R1)+i(R2)=0

Thecurrentthroughallresistorsinseriesisthe

same,sousingOhm’slaw:

V1=[(R1)/(R1+R2)]*V

V2=[(R2)/(R1+R2)]*V

V=V1+V2=i(R1+R2)

 Current Division (Parallel Resistors):

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Chapter 3

Nodal Analysis:Focuseson current flowingintoandoutofeachnode 

usingKCL.BecauseV=I/R,weareactuallygoingtofindthenode

voltagesintheend(v1,v2,v3,etc.).

Steps for Nodal Analysis:

1. Identifynodesinthecircuit(usecoloringmethodifnecessary)

2. ApplyKCLateachnode(exceptforthegroundnode)

3. SolvetheKCLequationsusingamatrixtofindtheunknown

nodevoltages.

Tips:

IFthereisavoltagesourcebetweentwononreferencenodes

(v1andv2),thatbecomesasupernode.Treatthesupernodeas

1node,andwriteaconstraintequationusingKCL(current

comingin=currentflowingout,ANDvoltage amount = v1-v2,

usingthe+ontheVsasthepositivenodevoltageand-onthe

Vsasthenegativenodevoltage).

Ifthereisavoltagesource(Vs)betweenanonreferencenode

(V1)andareferencenode(Ground),do[(Vs-V1)/resistance]

IfthereisaVoltageSource(Vs)rightnexttoanonreference

node(v1)anditlookslikeitisthesamevoltage,itprobablyis!

Do Nodal Analysis If the circuit contains:

Manyelementsinparallel

CurrentSources

Supernodes

CircuitswithfewerNodesthanmeshes

IftheNodeVoltageiswhatisbeingsolvedfor

Non-PlanarCircuits

Mesh Analysis:UsesKVLtofindunknown"MeshCurrents"(only

appliesinplanarcircuits).

Steps for Mesh Analysis:

1. Assignmeshcurrentstomeshes

2. ApplyKVLtoeachofthemeshes(loops)

3. Solvetheresultingequationstogetthemeshcurrents

Tips:

ThemeshcurrentsuseKVLcalculatedCLOCKWISEaroundthe

mesh/loop.

IfthemeshcurrentgoesWITHtheVoltagesorce(from-->(-+)

-->),thenthevoltagesourceshouldbenegativeintheKVL

equation.

IfthemeshcurrentgoesAGAINSTtheVoltagesource(from-->

(+-)-->),thenthevoltagesourceshouldbepositiveintheKVL

equation.

IFthereisaresistor bordering twomeshes ,makethe1stmesh

currentyouareworkingonpositiveandtheborderingmesh

currentnegative.

Ifthereisacurrent source isolated toamesh ,thatmesh

current=thecurrentsource.

Ifthereisacurrent source bordering twomeshes ,thatis

classifiedasaSUPERMESH.Fortheconstraintequation,use

themeshcurrentsborderingthecurrentsourceandtheir

directiontodeterminewhichispositiveandwhichisnegative.

Ifameshcurrentisgoingthesamedirectionasthecurrent

source,thenitispositiveintheconstraintequation.Ifamesh

currentisgoingagainstthecurrentsource,thenitisnegativein

theconstraintequation.

AftertheconstraintequationiswrittenfortheSUPERMESH,

youthenREMOVE(createanopencircuit)thewirecontaining

thecurrentsourceandresistoritisattachedto.Then,apply

KVLontheremainingMesh,usingI1andI2accordingly

(whateverresistorsappliedtoI1beforethesupermeshbranch

wasremoved,stillappliestoI1,andviceversaforI2).

Do Mesh Analysis if the circuit contains:

Manyelementsinseries

Voltagesources

Supermeshes

Acircuitwithfewermeshesthannodes

Ifabranch/meshcurrentiswhatisbeingsolvedfor

Chapter 4

Linearity:AsVoltagegoesup,Currentgoesup

proportionally.Theresponseofacircuittoasum of

sourceswillbethesum of the individual responses

fromeach source separately.

Linearity:

V=iR

k(iR)=k(V)

V=(i1+i2)R=(i1)R+(i2)R=V1+V2

Superposition:Iftherearetwo or more independent sourcesthereareThree waysto

solveforthecircuitparameters:NodalAnalysis,MeshAnalysis,Superposition

SuperpositionPrinciplestatesthatthevoltage/Currentthroughanelementina

linearcircuitisthetotal sum of the voltages/currentsthroughthatelement

duetoeachindependentsourceactingalone.

Steps for Superposition:

1. Turnoffallindependentsourcesexceptonesource.Findtheoutput(voltageor

current)duetothatactivesourceusingtechniquesinChapter2/3

2. Repeatstep1foreachoftheotherindependentsources

3. Findthetotalcontribution(voltageorcurrent)byaddingallthecontributions

(voltagesorcurrents)duetoeachindependentsource.

TipsforSuperposition:

Toremoveacurrentsource,replaceitwithanopencircuit(i=0).

Toremoveavoltagesource,replaceitwithashortcircuit(v=0).



Source Transformation:ReplacingaVoltagesource(Vs)inseries

witharesistorRbyacurrentsource(Is)inparallelwitharesistorR,

orviceversa.

Vs=Is*R

Is=Vs/R

Thecurrentsourceisdirectedtowardsthepositiveterminal

ofthevoltagesource

SourceTransformationisnotpossiblewhenR=0orR=∞

Thevenin's Theorem (Thevenin Equivalents):Whenyouhaveone

variableelementthatyouaretryingtoanalyze,calledtheload.

TheveninVoltage(Vth/Voc)=OpenCircuitVoltage(R=infinite)

TheveninResistance(Rth)=Resistancelookingintoterminalsa

andbwithall independent sources turned off.

TheveninCurrent(Ith/In/Isc)=Vth/Rth=Ith

Useavoltagesourceinserieswitharesistancetoreplace

complicatedlinearcircuits

Itispossiblefortheresultofthisanalysistoendupwithanegative

resistance.Thisimpliesthecircuitissupplyingpower.Thisisreasonable

withdependentsources

Norton's Theorem (Norton Equivalents):SimilartoThevenin’s

theorem,Norton’stheoremstatesthatalineartwoterminalcircuit

maybereplacedwithanequivalentcircuitcontainingaresistorand

acurrentsource.

NortonCurrent(In)=ShortCircuitCurrent(Isc)

NortonResistance(Rn)=TheveninResistance(Rth)

NortonVoltage(Vn)=[Vn=In*Rn]

TheNortonCurrent(In)isfoundbyshortcircuitingthe

circuit'sterminalsandmeasuringtheresultingcurrent.

NortoncurrentisrelatedtoTheveninvoltageandresistance

bysourcetransformation:In=Vth/Rth

Norton vs. Thevenin

Maximum Power Transfer:

P=[(Vth)^2/4(Rth)]



Chapter 5 Concepts

5TerminalsfoundonallOp-amps:

a. Invertinginput

b. Non-invertinginput

c. Output

d. Positiveandnegativepowersupplies

Outputvoltageofanop-ampcanbefoundusingthefollowingequation:Vo=A*Vd=A*(V+-V-)

oAistheOpenLoopGain,whichisdifferentfromtheclosedloopgain.

Feedback:Theoutputoftheop-ampisfedbacktotheinvertingterminal,givingtheop-amp

"NegativeFeedback".

VoltageSaturation:Theoutputvoltageoftheop-ampcannotexceedtheinputvoltages.Therefore,

whenanoutputvoltageshouldexceedthepossiblevoltagerange,theoutputremainsateitherthe

minimumormaximumsupplyvoltage.

Ideal Op-Amps:

oInfiniteOpenLoopGain(A)

oInfiniteInputResistance(Ri)

oZerooutputResistance(Ro)

oZeroinputcurrenttotheinverting/noninvertingterminals(Io)

oOutputcurrentisNOTzero

Inputvoltagesontheinverting/noninvertingterminalsareequal

InvertingOp-amp(ideal)

IdealOp-AmpRulesapply

Vo=(-Rf/R1)*Vi

Equivalent circuit:

Non-inverting Op-Amp (Ideal)

IdealOp-AmpRulesapply

Vo=Vi*[1+(Rf/R1)]

Voltage Follower (Ideal, non-inverting op-amp)

Vi=Vo

Summing Op-Amp

Usestheinvertingamplifierandseveralinputs(eachwiththeir

ownresistor),thesummingamplifiercanbeusedtocreatea

simpledigitaltoanalogconverter(DAC).



I1=[(V1-Va)/R1]

I2=[(V2-Va)/R2]

I3=[(V3-Va)/R3]



Ia=I1+I2+I3

Ia=[(Va-Vo)/Rf]



Vo=-[(Rf/R1)*V1+(Rf/R2)*V2+(Rf/R3)*V3]

Difference Amplifier

-Voisproportionaltothedifferencebetweenthetwoinputs.

-Va=Vbduetonegativefeedback.

CommonModeRejection:Adifferenceamplifierrejectsanysignal

thatiscommontothetwoinputs,followingthisequation:

Instrumentation Amplifier

-Places2non-invertingamplifiersbeforethe

differenceamplifiertoincreasetheimpedance

ofthedifferenceamplifier

Ifthisconditionistrueforadifferenceop-amp,theoutputisthen:

Chapter 6 Concepts

Capacitors

oCapacitorFormulas:

CapacitanceFormula:C=(E*A)/d

E=permittivityofdialectric

A=crosssectionalareaofplates

D=distancebetweenplates

Chargestoredinacapacitor isequaltothecapacitance*Voltage:q =

C*V

Takethederivativeofthisequationtofindcurrentinacapacitor:I =

C*(dv/dt)

Thisisalsothe"ICE"partofthe"ELITHEICEMAN"acronym

Tofindvoltageinacapacitorduetocurrent:







oCapacitorProperties:

WhenthevoltageisNOTchanging(constant),thecurrentthroughthe

capacitoriszero(opencircuitatDCconditions).

VoltageonthecapacitorplatesCANNOTchangeinstantaneously.

Ifthevoltageonthecapacitordoesnotequaltheappliedvoltage,charge

willflowuntilthecapreachestheappliedvoltage.

Parallel Capacitors: Actlikeseriesresistors,orlikeconductancevalues.

C1+C2+C3=Ceq

Series Capacitors: ActlikeParallelResistors,orlikeconductancevalues.

[(1/C1)+(1/C2)+(1/C3)]^-1=Ceq

Inductors

oInductorformulas:

Forasolenoid,theinductanceformulais:L=(N^2*u*A)/l

U=permeabilityofthecorematerial

N=numberofturnsofthewire

A=crosssectionalarea

L=length

Ifcurrentispassedthroughaninductor,thevoltageacrossitisdirectly

proportionaltotherateofchangeofcurrentflowingthroughit:V =

L*(di/dt)

Thisisthe"ELI"partofthe"ELITHEICEMAN"acronym

Thecurrentstoredinaninductorduetovoltage:







oInductorProperties:

IfthecurrentacrossaninductorisNOTchanging(constant),thevoltage

acrossaninductoriszero(shortcircuitatDCconditions).

ThecurrentthroughaninductorCANNOTchangeinstantaneously.

Series Inductors: Justlikeseriesresistors.L1+L2+L3=Leq

Parallel Inductors: Justlikeparallelresistors.[(1/L1)+(1/L2)+(1/L3)]^-

1=Leq

CombiningOp-amps,Capacitors,andinductors:

Integrator

Differentiator

EquationtofindVo:



Chapter 7 Concepts:

First-OrderCircuits(RCandRLCircuits)

CircuitExcitation

NoIndependentSource(NaturalResponse)

oaDCsourceissuddenlydisconnected

oStoredenergyintheCapacitor/Inductoristhen

releasedtothesystemoveraperiodoftime

WithindependentSource(ForcedResponse)

oDC,exponential,sinusoidal.Energyprovidedby

sources

NaturalResponse+ForcedResponse=CompleteResponse

Source Free RC Circuits

Sincewecanassumethecapacitorwasinitiallycharged,att=0theinitial

voltagethroughthecapacitoris:

V(0)=0

Asthecapacitorbeginstodischarge,thenaturalresponsestartstooccur

andvoltagestartstodissipate:

Timeconstant(thespeedatwhichthevoltagedropsto1/eofthe

initialvoltage):

Step response of RC Circuits

WhenaDCsourceissuddenlyappliedtoaRCcircuit,thesourcecanbe

modeledasastepfunction.

Thecompleteresponse:



Thefirstpartisthenaturalresponseofthecapacitor/inductordue

totheenergystoredinit.

Thesecondpartistheforcedresponseduetotheconnected

voltagesource.Vs = V(∞)andVo = V(0)

OnceV(t)isfound,currentcanbefound:

i(t)=

RL Circuits

oSourceFreeRLCircuits

Currentcannotchangeinstantaneously

Thekeytoworkingwiththistypeofsituationis:

Startwiththeinitialvoltageacrossthecapacitorandthetime

constantRC.

Withthesetwoitems,thevoltageasafunctionoftimecanbe

known.

Fromthevoltage,thecurrentcanbeknownbyusingthe

resistanceandOhm’slaw.

TheresistanceofthecircuitisoftentheTheveninequivalent

resistance.

Singularity Functions:

Unit Step Function

Theswitchingtimecanbeshiftedto

t=toby:

Unit Impulse Function

Thederivativeoftheunitstep

function

Unit Ramp Function

IntegrationoftheUnitStepFunction

Wearelookingforthecurrentthroughtheinductor

Therefore,wemustdetermineitsvalueasafunctionoftime

Initialcurrentpassingthroughtheinductoratt=0:

I(0)=0

Astheinductorbeginstoreleaseenergyintothesystem,thenatural

responsestartstooccur:

Timeconstant:

oStepResponseofRLCircuits

Steady-statecurrentthroughaninductor:

Currentcannotchangeinstantaneouslythroughaninductor:

i(0+)=I(0-)=Io

Completeresponseofcurrentthroughaninductor:





Chapter 8 Concepts

Second-orderCircuits(RLCcircuits)

oStartbygettinginitialconditionsI(0)anddI(0)/dt(forparallelRLC

circuits,V(0)anddV(0)/dt):

Capacitor:OpenCircuitatlong-termconditions.Voltage

cannotchangeabruptly

V(0-)=V(0+)=Vo

Inductor:ShortCircuitatlong-termconditions.Current

cannotchangeabruptly.

i(0-)=I(0+)=Io

oFigureoutwhichdampingcasethecircuitwillrequire,solve

characteristiceqn.forroots,applythemincorrectformula,solveforI

orV:

Series RLC Circuits:

Parallel RLC Circuits:

AllthesameequationsasseriesRLC,except:

ChangeI(t)-->V(t)

Vs=V(∞)

Change:

Damping Responses:

Overdamp

ed(the

rootsare

realand

negative)

Critically

Damped

(rootsare

realand

equal)

Underdam

ped(roots

are

complex)



oDampingfactor,UndampenedNaturalFrequency,damped

naturalFrequency(shownbelowinorder)

oSolutionstocharacteristicequation:

oAlsocanbefoundusingquadraticformula:

oUnderdampedcaseroots:

Overdamped

Critically

Damped

Underdamp

Chapter 9

SinusoidalVoltage:V(t)=Vmsin(ωt± ɸ)

oThefunctionrepeatseveryperiod,oreveryTseconds

oT=2π/ω

oFrequency(Hertz):f =1/T

oAngularfrequency:ω=2πf

Complexnumbers:

oRectangularform:z=x+jy

oPolarform:Z=r<ɸ

oExponentialform:Z=rejɸ

Rectangulartopolar:

r=sqrt(x2+y2)

ɸ=tan-1(y/x)

Polartorectangular:

Chapter 10

StepstoanalyzeACCircuits:

oTransformthecircuittothephasororfrequency

domain

oSolvetheproblemusingcircuittechniques

oTransformbacktotimedomain.

PerformthefollowingasinDC:

oNodalAnalysis

oMeshAnalysis

oSuperposition

oSourceTransformation

oTheveninandNortonEquivalents

oOpAmpAnalysis

X=rcosɸ

Y=rsinɸ

Phasors

oConvertingfromtimedomaintophasor:

V(t)=Vmcos(ωt± ɸ)-->Vm<ɸ

V(t)=Vmsin(ωt± ɸ)-->Vm<ɸ-90°

Currentsworkthesameway,justreplaceVmwithIm.

oPhasorRelationships(ELITHEICEMAN)

Resistors:V=IR

VoltageandCurrentareinphasewitheach

other(0°difference)

Inductors:R=V/I=jωL

ELI-->VoltageLeadsCurrentby90°,orcurrent

lagsvoltageby90°

Capacitors:R=-j/ωC

ICE-->Currentleadsvoltageby90°

oImpedance-->phasorversionofresistance

oAdmittance-->inverseofimpedance,phasorversionof

conductance

oIntherectangularformofImpedance(z=x+jy),thereal

part(x)istheresistance,andtheimaginarypartisthe

reactance(jy)

Whentheimpedanceispositive,it'sinductive(jwl),

whenit'snegativeit'scapacitive(-j/ωC)

oIntherectangularformofadmittance(z=x+jy),thereal

part(x)istheconductance,andtheimaginarypartisthe

susceptance(jy)

Chapter 11

Instantaneous Power :(1/2)VmIm[cos(ɸv-ɸi)+cos(2ωt+ɸv+ɸi)]

oTherateatwhichanelementabsorbspower,orthepoweratanyinstantintime

o1stcosineisconstantpower,2ndcosineissinusoidalpower

Average Power :

o(1/2)VmImcos(ɸv-ɸi)

o***Averagepowerabsorbedbyaninductorandcapacitoriszerowatts.***

oAveragepowerisalso:

Forperiodiccurrent:  

ForDCCurrent:

Resistivevs.ReactivePower

oResistive:Whenɸv-ɸi=0°,thevoltageandcurrentareinphaseandthecircuitispurelyresistive

P=(1/2)VmIm=(1/2)Im2R=(1/2)|I|2R

oReactive:Whenɸv-ɸi=±90°,thecircuitabsorbsnopowerandispurelyreactive

P=(1/2)VmImCos(±90°)=0

MaximumAveragePower

oPmax=(|VTh|2)/(8RTh)

RMS

oforasinusoidalwaveform,theRMSvalueisrelatedtotheamplitudeasfollows:

oRMSPowercanbedeterminesfromeitherRMScurrentorvoltage:

P=(IRMS)2R=(VRMS)2/R

Complex Power

oS=(1/2)VI*

I*=complexconjugateofcurrent

oS=P+jQ

P=Real power -->

Scos

(

θv−θi

)

jQ=Reactive power-->

Ssin

(

θv−θi

)

oS=VRMSIRMS*

oS=|VRMS||IRMS|<(ɸv-ɸi)

Apparent power

oTheproductofRMSvoltageandcurrentwillbecalledapparentpower.

|S|=|VRMS||IRMS|=

√

P2+Q2

Power Factor

oP/S=cos(ɸv-ɸi)-->cos-1(P/S)=PowerFactorRatio(between0and1)

P=RealPower

S=Apparentpower

RealPower(P)=ApparentPower(S)*PowerFactor(PF)

PowerFactor=cos(ɸv-ɸi)



Addingacapacitor

oTomitigatetheinductiveaspectoftheload,acapacitorisaddedinparallelwiththeload.

oWiththesamesuppliedvoltage,thecurrentdrawislessbyaddingthecapacitor.

C=Qc

ωV rms

2=P

(

tan θ1−tan θ2

)

ωV rms

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