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Berkeley ELENG 100 - Lecture Notes

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Slide 1EE100 Summer 2008 Bharathwaj MuthuswamyEE100Su08 Lecture #11 (July 21st2008)• Bureaucratic Stuff– Lecture videos should be up by tonight– HW #2: Pick up from office hours today, will leave them in lab.REGRADE DEADLINE: Monday, July 28th2008, 5:00 pm PST, Bart’s office hours.– HW #1: Pick up from lab.– Midterm #1: Pick up from me in OHREGRADE DEADLINE: Wednesday, July 23rd2008, 5:00 pm PST. Midterm: drop off in hw box with a note attached on the first page explaining your request. •OUTLINE– QUESTIONS?– Op-amp MultiSim example– Introduction and Motivation– Arithmetic with Complex Numbers (Appendix B in your book)– Phasors as notation for Sinusoids– Complex impedances – Circuit analysis using complex impedances– Derivative/Integration as multiplication/division– Phasor Relationship for Circuit Elements– Frequency Response and Bode plots• Reading– Chapter 9 from your book (skip 9.10, 9.11 (duh)), Appendix E* (skip second-order resonance bode plots)– Chapter 1 from your reader (skip second-order resonance bode plots)Slide 2EE100 Summer 2008 Bharathwaj MuthuswamyOp-amps: Conclusion• Questions?• MultiSim ExampleSlide 3EE100 Summer 2008 Bharathwaj MuthuswamyTypes of Circuit ExcitationLinear Time-InvariantCircuitSteady-State ExcitationLinear Time-InvariantCircuitORLinear Time-InvariantCircuitDigitalPulseSourceTransient ExcitationLinear Time-InvariantCircuitSinusoidal (Single-Frequency) ExcitationÆAC Steady-State(DC Steady-State)Slide 4EE100 Summer 2008 Bharathwaj MuthuswamyWhy is Single-Frequency Excitation Important?• Some circuits are driven by a single-frequency sinusoidal source. • Some circuits are driven by sinusoidal sources whose frequency changes slowly over time.• You can express any periodic electrical signal as a sum of single-frequency sinusoids – so you can analyze the response of the (linear, time-invariant) circuit to each individual frequency component and then sum the responses to get the total response.• This is known as Fourier Transform and is tremendously important to all kinds of engineering disciplines!Slide 5EE100 Summer 2008 Bharathwaj MuthuswamyabcdsignalsignalTi me (ms)Frequency (Hz)Signal (V)Relative AmplitudeSignal (V)Signal (V) Representing a Square Wave as a Sum of Sinusoids(a)Square wave with 1-second period. (b) Fundamental component (dotted) with 1-second period, third-harmonic (solid black) with1/3-second period, and their sum (blue). (c) Sum of first ten components. (d) Spectrum with 20 terms.Slide 6EE100 Summer 2008 Bharathwaj MuthuswamySteady-State Sinusoidal Analysis• Also known as AC steady-state• Any steady state voltage or current in a linear circuit with a sinusoidal source is a sinusoid.– This is a consequence of the nature of particular solutions for sinusoidal forcing functions.• All AC steady state voltages and currents have the same frequency as the source.• In order to find a steady state voltage or current, all we need to know is its magnitude and its phase relative to the source – We already know its frequency.• Usually, an AC steady state voltage or current is given by the particular solution to a differential equation.Slide 7EE100 Summer 2008 Bharathwaj MuthuswamyExample: 1storder RC Circuit with sinusoidal excitationR+-CVst=0Slide 8EE100 Summer 2008 Bharathwaj MuthuswamySinusoidal Sources Create Too Much Algebra)cos()sin()( wtBwtAtxP+=)cos()sin()()( wtFwtFdttdxtxBAPP+=+τ)cos()sin())cos()sin(())cos()sin(( wtFwtFdtwtBwtAdwtBwtABA+=+++τGuess a solutionEquation holds for all time and time variations are independent and thus each time variation coefficient is individually zero0)cos()()sin()(=−++−− wtFABwtFBABAττ0)(=−+BFABτ0)(=−−AFBAτ12++=ττBAFFA12+−−=ττBAFFBTwo terms to be generalPhasors (vectors that rotate in the complex plane) are a clever alternative.Slide 9EE100 Summer 2008 Bharathwaj MuthuswamyComplex Numbers (1)• x is the real part• y is the imaginary part• z is the magnitude• θ is the phase(1)j=−θzxyreal axisimaginary axis• Rectangular Coordinates Z = x + jy• Polar Coordinates: Z = z ∠θ• Exponential Form: θcoszx=θsinzy=22yxz +=xy1tan−=θ(cos sin )zjθθ=+Zjjezeθθ==ZZ0211 101190jjejeπ==∠°==∠ °Slide 10EE100 Summer 2008 Bharathwaj MuthuswamyComplex Numbers (2)22cos2sin2cos sincos sin 1jjjjjjeeeejejeθθθθθθθθθθθθ−−+=−==+=+=jjezezθθθ===∠ZZEuler’s IdentitiesExponential Form of a complex numberSlide 11EE100 Summer 2008 Bharathwaj MuthuswamyArithmetic With Complex Numbers• To compute phasor voltages and currents, we need to be able to perform computation with complex numbers.– Addition– Subtraction– Multiplication– Division• Later use multiplication by jω to replace:– Differentiation– IntegrationSlide 12EE100 Summer 2008 Bharathwaj MuthuswamyAddition• Addition is most easily performed in rectangular coordinates:A = x + jyB = z + jwA + B = (x + z) + j(y + w)Slide 13EE100 Summer 2008 Bharathwaj MuthuswamyAdditionReal AxisImaginary AxisABA + BSlide 14EE100 Summer 2008 Bharathwaj MuthuswamySubtraction• Subtraction is most easily performed in rectangular coordinates:A = x + jyB = z + jwA - B = (x - z) + j(y - w)Slide 15EE100 Summer 2008 Bharathwaj MuthuswamySubtractionReal AxisImaginary AxisABA - BSlide 16EE100 Summer 2008 Bharathwaj MuthuswamyMultiplication• Multiplication is most easily performed in polar coordinates:A = AM∠θB = BM∠φA × B = (AM × BM) ∠ (θ + φ)Slide 17EE100 Summer 2008 Bharathwaj MuthuswamyMultiplicationReal AxisImaginary AxisABA × BSlide 18EE100 Summer 2008 Bharathwaj MuthuswamyDivision• Division is most easily performed in polar coordinates:A = AM∠θB = BM∠φA / B = (AM/ BM) ∠ (θ − φ)Slide 19EE100 Summer 2008 Bharathwaj MuthuswamyDivisionReal AxisImaginary AxisABA / BSlide 20EE100 Summer 2008 Bharathwaj MuthuswamyArithmetic Operations of Complex Numbers• Add and Subtract: it is easiest to do this in rectangular format– Add/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 phases1212111111122 222 2 2 22 1 12 2 1 12 22 1 12 2 1 12 2()212 12 12212cos sincos sin(cos cos) (sin sin )(cos cos) (sin sin )() ()()/(/)jjjze z z jzze z z jzzz jzzzz jzzzze zzzzeθθθθθθ θθθ θθθθθθθθθθθ+==∠= +==∠= ++= + + +−= − + −×=× =×∠+=11111ZZZZZZZZZZ12()12 1 2(/)(


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Berkeley ELENG 100 - Lecture Notes

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