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Berkeley ELENG 105 - Lecture 17

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R. T. HoweEECS 105 Spring 2002 Lecture 17Dept. of EECSUniversity of California, BerkeleyLecture 17• Last time:– Wrap-up two-port MOS amplifiers• Today :– Sinusoidal signals– Phasor representation of sinusoidsR. T. HoweEECS 105 Spring 2002 Lecture 17Dept. of EECSUniversity of California, BerkeleySinusoidal Function Reviewvt() v ωt φ+()cos=amplitudefrequencyphase (degrees(radian) ... ω = 2π f = 2π (1/T)(half ofpeak-to-peak)or radians)R. T. HoweEECS 105 Spring 2002 Lecture 17Dept. of EECSUniversity of California, BerkeleyGraphical Descriptionv1t() v ωt()cos=v2t() v ωt 45–()cos=ω2πT------=TR. T. HoweEECS 105 Spring 2002 Lecture 17Dept. of EECSUniversity of California, BerkeleyWhy are Sinusoids Important?• Any periodic signal v(t) can be expressed as a sum of sinusoidal signals by a Fourier series expansion (EECS 20N, EE 120)• The response of a linear circuit to a sinusoidal input, as a function of its frequency ω, leads to insights into the behavior of the circuit.R. T. HoweEECS 105 Spring 2002 Lecture 17Dept. of EECSUniversity of California, BerkeleyLinear Circuits• Theorem: solutions for voltages and currents in a linear circuit (i.e., one consisting of R, L, C and dependent sources Gm, Rm, Av, and Ai) with a sinusoidal signal as the input are:R. T. HoweEECS 105 Spring 2002 Lecture 17Dept. of EECSUniversity of California, BerkeleyRC Circuit with Sinusoidal Input Rvc(t)C iR iCvs(t) + - vs(t)= Vs cos(ωt) : set phase of source to zero (use as the reference)vc(t)= Vc cos(ωt + φ) : solution is a sinusoidal signal with the same frequency, but with a different amplitude and phase-shifted with respect to the sourceR. T. HoweEECS 105 Spring 2002 Lecture 17Dept. of EECSUniversity of California, BerkeleyA Better Technique• It is much more efficient to work with imaginary exponentials as “representing”sinusoids, since these functions are direct solutions of linear differential equations:ddt----- ejωt()jω ejωt()=• Note that EEs use j = (-1)1/2rather than i, since thesymbol i is already taken for currentR. T. HoweEECS 105 Spring 2002 Lecture 17Dept. of EECSUniversity of California, BerkeleyUsing Imaginary ExponentialssccvvdtdvRC =+Substitute:tjssevtvω=)()()(φω+=tjccevtvResult:tjstjctjcevevevjωφωφωωτ=+++)()()(R. T. HoweEECS 105 Spring 2002 Lecture 17Dept. of EECSUniversity of California, BerkeleyFinding the Amplitude Ratiotjstjjcjceveevevjωωφφωτ=+ ])([scjjvveej =+ ][φφωτ)1(1ωτωτφφφjeeejvvjjjsc+=+=−AmplitudeRatio:Answer is a real number, so take magnitude… use to find amplitude and phase()211ωτ+=scvvR. T. HoweEECS 105 Spring 2002 Lecture 17Dept. of EECSUniversity of California, BerkeleyGraphical Result for Amplitude Ratio ω 1/τ 10/τ 1/10τ 1.0 0.5 0.707R. T. HoweEECS 105 Spring 2002 Lecture 17Dept. of EECSUniversity of California, BerkeleyAmplitude: a new representation• We are interested in very small ratios (e.g., Vc/Vs= 0.0001)• Therefore, we use a log plot … but we also define a new function called the deciBel (after Alex. Graham Bell)(Vc/Vs)dB= 20 log10(Vc/Vs)• Examples: Vc/Vs= 0.0001 Æ (Vc/Vs)dB= -80 dBVc/Vs= 0.707 Æ (Vc/Vs)dB= -3 dBR. T. HoweEECS 105 Spring 2002 Lecture 17Dept. of EECSUniversity of California, BerkeleyFinding the PhaseCollect real and imaginary parts; latter must be zero:Use Euler’s formula to convert to rectangular form:csjjvveej /][ =+φφωτ(a real number)csvvjjj/)sin(cos)sin(cos=+++φφφφωτ0sincos)Im(=+=⋅φφωτωτφ−=tanR. T. HoweEECS 105 Spring 2002 Lecture 17Dept. of EECSUniversity of California, BerkeleyGraphical Result for Phase φ ω 1/τ 10/τ 1/10τ 0 -45 -90R. T. HoweEECS 105 Spring 2002 Lecture 17Dept. of EECSUniversity of California, BerkeleyFinding the “Real” Waveform• How to connect the imaginary exponential solution to the measured waveform v(t)? Conventionally, v(t) is the real part of the of the imaginary exponential)cos()Re()(φωφω+=+tvvetjR. T. HoweEECS 105 Spring 2002 Lecture 17Dept. of EECSUniversity of California, BerkeleyPushing This Idea Further …There are two parameters needed to define a sinusoidal signal:* magnitude* phaseWhy not work with a complex number as the signal and eliminatethe imaginary exponential from the analysis (it cancelled out)?Define the complex number consisting of the amplitude and phasea sinusoidal signal as a phasortjVetvtvtvωφω=⇔+=


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Berkeley ELENG 105 - Lecture 17

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