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Berkeley ELENG 40 - Complex Numbers and Phasors

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EE40 Lec 10EE40 Lec 10Complex Numbers and PhasorsComplex Numbers and PhasorsProf Nathan CheungProf. Nathan Cheung09/29/2009Reading: Hambley Chapter 5 and Appendix ASlide 1EE40 Fall 2009 Prof. CheungHambley Chapter 5 and Appendix AOUTLINE• Phasors as notation for Sinusoids•Arithmetic with Complex NumbersArithmetic with Complex Numbers• Complex impedances •Circuit analysis using complex impedances•Circuit analysis using complex impedances• Dervative/Integration as multiplication/division•Phasor Relationship for Circuit Elements•Phasor Relationship for Circuit ElementsSlide 2EE40 Fall 2009 Prof. CheungTypes of Circuit ExcitationLinear Time-IitLinear Time-IitInvariantCircuitInvariantCircuitSteady-State Excitation ORLinear TimeDigital(DC Steady-State)Step ExcitationLinear Time-InvariantCircuitDigitalPulseSourceLinear Time-InvariantCi itCircuitTransient ExcitationsCircuitSinusoidal (Single-Frequency) ExcitationSlide 3EE40 Fall 2009 Prof. CheungTransient ExcitationsFrequency) ExcitationÆAC Steady-StateSinusoids• Amplitude: VM()() cosMvt V tωθ=+• Angular frequency: ω = 2π f – Radians/sec • Phase angle: θ• Frequency: f = 1/T– Unit: 1/sec or Hz• Period: T– Time necessary to go through one cycleSlide 4EE40 Fall 2009 Prof. CheungSinusoidsWhat is the amplitude, period, frequency,d di f f thi i id?and radian frequency of this sinusoid?6824622000.010.020.030.040.05-6-4-200.010.020.030.040.05Slide 5EE40 Fall 2009 Prof. Cheung-8-6Sinusoidal Sources Create Too Much Algebra)cos()sin()()( wtFwtFdttdxtxBAPP+=+τGuess a solution)cos()sin()( wtBwtAtxP+=)()i())cos()sin(())()i((FFwtBwtAdBA+Guess a solution)cos()sin())()(())cos()sin((wtFwtFdtwtBwtABA+=++τ0)cos()()sin()(=−++−− wtFABwtFBABAττ0)(FBAτ0)(=−+BFABτ0)(=−−AFBAτ+τBAFF−τFF12++=ττBAFFA12+−=ττBAFFBPhasors (vectors that rotate in the complex Slide 6EE40 Fall 2009 Prof. Cheungso s (vec o s o e e co p eplane) are a clever alternative.Complex Numbers (1)ith l tii•xis the real part• y is the imaginary partih id(1)jyimaginary axis•zis the magnitude• θ is the phase(1)j=−θrealxreal axis• Rectangular Coordinates θcoszx=θsinzy=22yxz+=y1tan−=θZ = x + jy• Polar Coordinates: yxz+=xtan=θ(cos sin )zjθθ=+ZZ = z ∠θ• Exponential Form: jjθθZZ011 10jjeπ==∠°Slide 7EE40 Fall 2009 Prof. Cheungjjezeθθ==ZZ21190jje==∠ °Complex Numbers (2)cos2jjeeθθθ−+=Euler’s Identities2sin2jjeejθθθ−−=222cos sinjjjejθθθθ=+22cos sin 1jeθθθ=+=jjezezθθθ===∠ZZExponential Form of a complex numberSlide 8EE40 Fall 2009 Prof. CheungeeθArithmetic With Complex Numbers• To compute phasor voltages and currents, we need to be able to perform computation with lbcomplex numbers.– AdditionSbt ti–Subtraction– Multiplication–Division• (And later use multiplication by jω to replace – Differentiation–IntegrationSlide 9EE40 Fall 2009 Prof. CheunggAddition• Addition is most easily performed in rectangular coordinates:A = x + jyB = z + jwA+ B = (x + z) + j(y + w)Imaginary gyAxisA + BReal ABSlide 10EE40 Fall 2009 Prof. CheungAxisSubtraction• Subtraction is most easily performed in rectangular coordinates:A = x + jyB = z + jwA- B = (x -z) + j(y -w)Imaginary AiAxisReal ABSlide 11EE40 Fall 2009 Prof. CheungAxisA - BMultiplication• Multiplication is most easily performed in polar coordinates:A = AM∠θB = BM∠φA×B = (AM ×BM) ∠(θ + φ)IiImaginary AxisBA × BReal AiASlide 12EE40 Fall 2009 Prof. CheungAxisDivision• Division is most easily performed in polar coordinates:A = AM∠θB = BM∠φA/ B = (AM/ BM) ∠(θ −φ)Imaginary AxisBReal AxisAA / BSlide 13EE40 Fall 2009 Prof. CheungArithmetic Operations of Complex Numbers• Add and Subtract: it is easiest to do this in rectangular format–Add/subtract the real and imaginary parts separatelyAdd/subtract the real and imaginary parts separately• Multiply and Divide: it is easiest to do this in exponential/polar format– Multiply (divide) the magnitudes– Add (subtract) the phases1cos sinjze z z jzθθθ θ==∠= +Z2111111122 222 2 2 22 1 12 2 1 12 2cos sincos sin(cos cos ) (sin sin )jze z z jzze z z jzzz jzzθθθ θθθ θθθ θθ==∠= +==∠= ++=++ +11ZZZZ122 1 12 2 1 12 22 1 12 2 1 12 2()212 12 12(cos cos ) (sin sin )(cos cos ) (sin sin )() ()( )jzz jzzzz jzzzze zzθθθθ θθθθθθθθ+++++−= − + −×=× =×∠+111ZZZZZZSlide 14EE40 Fall 2009 Prof. Cheung212 12 12212() ()( )/(/)zze=11ZZ12()12 1 2(/ )( )jzzθθθθ−=∠−Summary of Complex NumbersSlide 15EE40 Fall 2009 Prof. CheungPhasor: Rotating Complex Vector{})(tjjwtjeeVetVtvωφφωVReRe)cos()( ==+=Imaginary AxisRotates at uniformPhasorRotates at uniform angular velocity ωtReal VAxiscos(ωt+φ)φSlide 16EE40 Fall 2009 Prof. CheungThe head start angle is φ.Phasors and Complex Exponentials• A sinusoid can be described using a complex exponential:ejωt= cos ωt + j sin ωtSoSo,v(t) = VMcos (ωt + θ) = VM Re[ej (ωt + θ)]• The phasor of v(t) is given by:V = VM∠ θSlide 17EE40 Fall 2009 Prof. CheungSinusoids, Complex Exponentials, Phasors•Sinusoid:Sinusoid:z cos (ωt+θ)Complex exponential:•Complex exponential:Aejωt = z ej(ωt+θ)• Phasor:V= z∠θθSlide 18EE40 Fall 2009 Prof. CheungSinusoids <--> PhasorsSlide 19EE40 Fall 2009 Prof. CheungComplex Exponentials• Complex Exponentials provide the link between time functions and phasors–provide the link between time functions and phasors.– Allow dervatives and integrals to be replaced by multiplying or dividing by jω– make solving for AC steady state simple algebra with complex numbers.• Phasors allow us to express current-voltage grelationships for inductors and capacitors much like we express the current-voltage relationship for a resistorSlide 20EE40 Fall 2009 Prof. Cheungfor a resistor.Complex Impedance• AC steady-state analysis using phasors allows us to express the relationship between current and voltage using a formula that looks likes Ohm’s law:V = IZ•Zis calledimpedanceZis called impedance.Slide 21EE40 Fall 2009 Prof. CheungI-V Relationship for a CapacitorCv(t)+i(t)tdvCti)()(=C()-dtCti)(Suppose that v(t) is a sinusoid:v(t) = Re{VMej(ωt+θ)}Findi(t).Slide 22EE40 Fall 2009 Prof. CheungFind i(t).Capacitor i(t)C(t+i(t)tdvCti)()(=Cv(t)-dtCti)(=() ()() () () ()() cos( )2()()jt jtjt jt jt jtVvt V t e edv t CV d CVCωθ ωθωθ ωθ ωθ ωθωθ+−+++ ++=+= +() () () ()() ()()()22sin( ) cos( )22jt jt jt jtjt jtdv t CV d


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Berkeley ELENG 40 - Complex Numbers and Phasors

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