1Chapter 3 Alternating Current Circuits I•AC Voltage and Current - Phasors•RMS Voltage and Current•Reactance and Impedance•High Pass and Low Pass filters•RLC Resonance Circuits2Waves and Phasorsy(t) = A cos(ωt-φ)ω=2πf where f=1/T(!12,i2)eiωt = (0,i)ωtxiy(!12,i2)Phasoreiωt =z =(x,iy) = |z|eiπ/2 = ( cos(π/2), i sin(π/2)= (0,i)z =(x,iy) = |z|eiπ/4 = ( cos(3π/4), isin(3π/4)=3Phase Lag in an AC CircuitIn a general AC circuit (RLC) we have to consider that the voltageand current may be out of phase due to the circuit elements. V(t) = Vo sin(ωt) ~ Vo eiωt I(t) = Io sin(ωt-φ) ~ Io ei(ωt-φ)Vo eiωtωtIo ei(ωt-φ)V(t) I(t) φ=πV(t) I(t) φ=π/24 VRMS( )2=1T| V (t ) |2dt0T!= V02Tsin2("t )!dt0T!=Vo2T1 + cos(2"t )2!dt0T!=Vo22+Vo22T12"cos(2"t )0TVRMS( )2==Vo22+Vo22T12"cos(2"T ) # 1( )=Vo22+Vo22T12"cos(2"2$") # 1%&'()*!=0! "## $##!!!!=!!!VRMS=Vo2IRMS( )2=1T| I (t ) |2dt0T!= Io22!!!!!!!!!!!!!!!IRMS=Io2PRMS( )2=1T| I (t )V (t ) |2dt0T!=I0V0Tsin2("t ++) sin2("t )dt0T!Average and RMS Voltage, Current, PowerVAVG= 1TV(t) dt0T! IAVG= 1TI(t) dt 0T! PAVG= 1TI(t)V(t) dt 0T!5Reactance, Impedance, and Phasors Consider the general RLC circuit with V (t) = V0ej!t, I(t) = I0ej(!t"#):V (t) = LdIdt + IR + 1CI(t)dt$V (t) = I(t) i!L + R +1i!C%&'()* Re ac tan ce orComplex Im pedance! "### $### Ohm ' s Law Complex (AC) FormIn any AC Circuit the Re sistive, Capacitive, and Inductive elementscan be replaces by their Complex Im pedances!ZR=! R!=! XL Re sistive Re ac tan ce!!!!!!!!!!1 = e+i ! 0#=!0ZL= i!L( )= +i! XL Inductive Re ac tan ce!!!!!!!!+i = e+i !+/2!!!!!!!#= ++/ 2ZC=1i1!C%&'()*= "i! XC !!!!!!Capacitive Re ac tan ce!!!!!"i = e"i !+/2!!!!!!!#= "+/ 26Magnitude and PhaseThe total Re ac tan ce can be written as Z = R + i !L "1!C #$%&'( corresponding to a complex number XRXLZThe magnitude (length) of Z is | Z | = R2+ (!L "1!C)2The phase angle between XR and Z #Z= tan"1 (y / x) = tan"1!L "1!CR cos(#) =R| Z |φXC7Simple RC!Circuit - Low pass Filter (1)ZRZC What is the output voltage V and phase ! looking across the capacitor ?VC=ZCZR+ ZC Vo ei"t=!#i /"CR # i /"C Vo ei"t $ voltage divider eq| VC| = VCVC* = (#i /"C)(+i /"C)(R # i /"C)(R + i /"C)Vo =1 /"CR2+ 1 /"C( )2Vo!=1 /"RC1 + 1 /"RC( )2VoPhase : VC= (#i /"CR # i /"C)=Z! "# $#(R + i /"CR + i /"C)=1! "# $#=1 /"2C2R2+ 1 /"C( )2%&'()*X! "### $#### iR /"CR2+ 1 /"C( )2%&'()*Y! "### $### +C= tan#1YX%&'()*= tan#1#R /"C1 /"2C2%&'()*= tan#1#"RC( )= cot#1#1"RC%&'()*= tan#11"RC%&'()*#,/ 2At!!"= 1 / RC!!!+C= tan#11( )#,/ 2 =,/ 4 #,/ 2 = #,/ 4φR1/ωCgain = VC/ V0ZRZCVCVORCV(t) VCVC8Simple RC!Circuit - Low pass Filter (2) Break!Frequency!occurs!when!!!!!break=1RC!!"!!| VC| =12Vo = 0.707 Vo| VC| =11 + !RC( )2Vo !!!! fbreak=12#RCbreak frequency! "## $## fbfbφ=−π/4ln(0.707)9Simple RC!Circuit - High Pass FilterZCV(t)ZRWhat is the output voltage VR and phase !R looking across the resistor ?VR=ZRZR+ ZC"#$%&' Vo!ej(t =R(R ) j /(C)!Vo!ej(t | VR| = VRVR* = R2(R ) j /(C)(R + j /(C)Vo =RR2+ 1 /(C( )2Vo =(RC1 +(RC( )2Vogain = VR/ VO=(RC1 +(RC( )2Phase : VR= R2R2+ 1 /(C( )2"#$%&'+ jR /(CR2+ 1 /(C( )2"#$%&'!!!!!*!!!R= tan)1YX"#$%&'= tan)11(RC"#$%&'At!!(= 1 / RC!!!!R= tan)11( )=+/ 4θR1/ωCCRV(t) Vout10Simple RC!Circuit - High Pass Filter (2) | VR| =!RC1 + !RC( )2Vo fbreak=12"RCbreak frequency! "## $## !!!!!!| VR| =12Vo = 0.707 Vofbfbφ=π/4ln(0.707)11Differentiator and Integrator!ConnectionRCV(t) VC=1C i !dtint egratorRC>>T1/T >>1/RCf >>1/RCf > fbreak! "# $#RV(t) VR= R dq / dtdifferentiatorRC<<T1/T <<1/RCf << 1/RCf < fbreak! "## $## fbreak fbreak12Band Pass or Notch FilterωHI= ωLO=13RLC Circuitdampingspringmass14Resonance Condition and Q-factorωοZL=ZC at ω0Q = |ZL|/R = |ZC|/R15Power in AC Circuits16Decibel ScaleDECIBEL SCALE When measuring power gain and voltage gain in an amplifier or circuit we often use the dcibel scale. Power P db = 10 log(Pout/Pin) Voltage V db = 20 log10(Vout /Vin) = 20 log10 (gain) Vdb at the Break Frequency At the breaking frequency the gain Vout/Vin drops by a factor of 1/ 2 . This is called the -3 db point. Can you jusify this rema rk? Sound Power• Near total silence - 0 dB• A whisper - 15 dB• Normal conversation - 60 dB• A lawnmower - 90 dB• A car horn - 110 dB• A rock concert or a jet engine - 120 dB• A gunshot or firecracker - 140