The Last LegThe End is Near!Slide 3So far we have consideredExample LR CircuitTime Dependent Result:Slide 7Variable Emf AppliedSinusoidal StuffSlide 10Different FrequenciesAt t=0, the charged capacitor is connected to the inductor. What would you expect to happen??Slide 13The Math Solution:New Feature of Circuits with L and CThe GraphNote – Power is delivered to our homes as an oscillating source (AC)Producing AC GeneratorThe Real WorldSlide 20Slide 21The Flux:OUTPUTAverage value of anything:Average ValueSo …Slide 27RMSUsually Written as:Example: What Is the RMS AVERAGE of the power delivered to the resistor in the circuit:PowerMore Power - DetailsResistive CircuitConsider this circuitSlide 35Alternating Current CircuitsPhase TermSlide 38Example: household voltageSlide 40Slide 41Slide 42Resistors in AC CircuitsCapacitors in AC CircuitsI Leads V??? What the **(&@ does that mean??Slide 46Inductors in AC CircuitsSlide 48Phasor DiagramsSlide 50Slide 51Slide 52Slide 53Slide 54Slide 55Slide 56The Last LegThe Last LegThe Last LegThe Last LegThe Ups and Downs of CircuitsThe Ups and Downs of CircuitsChapter 31Chapter 31The End is Near!•Examination #3 – Friday (April 15th)•Taxes Due – Friday (April 15th)•Watch for new WebAssigns•Final Exam is Wednesday, April 27th.•Grades will be submitted as quickly as possible.That’s Two Weeks. That’s Two Weeks. That’s Two Weeks.So far we have consideredDC CircuitsResistorsCapacitorsInductorsWe looked at Steady State DC behaviorsTransient DC behaviors.We have not looked at sources that varied with time.Example LR Circuiti0equationcapacitor theas form same0:0 drops voltageof sumdtdqRCqEdtdiLiRESteady SourceTime Dependent Result:RLeREiLRtconstant time)1(/RLVariable Emf AppliedemfSinusoidalDCSinusoidal Stuff )sin( tAemf“Angle”Phase AngleSame FrequencywithPHASE SHIFTDifferent FrequenciesAt t=0, the charged capacitor is connected to the inductor. What would you expect to happen??The Math Solution:LCNew Feature of Circuits with L and CThese circuits produce oscillations in the currents and voltagesWithout a resistance, the oscillations would continue in an un-driven circuit.With resistance, the current would eventually die out.The GraphNote – Power is delivered to our homes as an oscillating source (AC)Producing AC Generatorx x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x xx x x x x x x x x x x x x x x x x x x x x x xThe Real WorldAThe Flux:tARemfitBAemftBAbulbsinsincos- ABOUTPUT)sin(0tVVemfWHAT IS AVERAGE VALUE OF THE EMF ??Average value of anything:Area under the curve = area under in the average boxTTdttfThdttfTh00)(1)(ThAverage ValueTdttVTV0)(1 0sin100TdttVTVFor AC:So …Average value of current will be zero.Power is proportional to i2R and is ONLY dissipated in the resistor,The average value of i2 is NOT zero because it is always POSITIVEAverage Value0)(10TdttVTV2VVrmsRMS22)(22)2(21)2(1002020020020220VVVdSinVVtTdtTSinTTVVdttTSinTVtSinVVrmsrmsTrmsTrmsUsually Written as:22rmspe akpeakrmsVVVVExample: What Is the RMS AVERAGE of the power delivered to the resistor in the circuit:ER~PowertRVRtRVRitPtRVRVitVV22020200sin)sin()()sin()sin(More Power - DetailsRVVVRRVPRVdSinRVPtdtSinRVPdttSinTRVPtSinRVtSinRVPrmsTT200202020220022002202202202212121)(21)(12)(1Resistive CircuitWe apply an AC voltage to the circuit.Ohm’s Law AppliesConsider this circuitCURRENT ANDVOLTAGE IN PHASERemfiiReAlternating Current Circuits is the angular frequency (angular speed) [radians per second].Sometimes instead of  we use the frequency f [cycles per second] Frequency  f [cycles per second, or Hertz (Hz)]  2 fV = VP sin (t -v ) I = IP sin (t -I )An “AC” circuit is one in which the driving voltage andhence the current are sinusoidal in time.v2V(t)tVp-VpvV(t)tVp-VpV = VP sin (wt - v )Phase TermVp and Ip are the peak current and voltage. We also use the “root-mean-square” values: Vrms = Vp / and Irms=Ip /v and I are called phase differences (these determine whenV and I are zero). Usually we’re free to set v=0 (but not I).22Alternating Current CircuitsV = VP sin (t -v ) I = IP sin (t -I )vV(t)tVp-VpVrmsI/I(t)tIp-IpIrmsExample: household voltageIn the U.S., standard wiring supplies 120 V at 60 Hz. Write this in sinusoidal form, assuming V(t)=0 at t=0.Example: household voltageIn the U.S., standard wiring supplies 120 V at 60 Hz. Write this in sinusoidal form, assuming V(t)=0 at t=0.This 120 V is the RMS amplitude: so Vp=Vrms = 170 V.2Example: household voltageIn the U.S., standard wiring supplies 120 V at 60 Hz. Write this in sinusoidal form, assuming V(t)=0 at t=0.This 120 V is the RMS amplitude: so Vp=Vrms = 170 V.This 60 Hz is the frequency f: so =2f=377 s -1.2Example: household voltageIn the U.S., standard wiring supplies 120 V at 60 Hz. Write this in sinusoidal form, assuming V(t)=0 at t=0.This 120 V is the RMS amplitude: so Vp=Vrms = 170 V.This 60 Hz is the frequency f: so =2f=377 s -1.So V(t) = 170 sin(377t + v).Choose v=0 so that V(t)=0 at t=0: V(t) = 170 sin(377t).2Resistors in AC CircuitsER~EMF (and also voltage across resistor): V = VP sin (t)Hence by Ohm’s law, I=V/R: I = (VP /R) sin(t) = IP sin(t) (with IP=VP/R)V and I“In-phase”VtIThis looks like IP=VP/R for a resistor (except for the phase change). So we call Xc = 1/(C) the Capacitive ReactanceCapacitors in AC CircuitsE~CStart from: