EECS 105 Spring 2002 Lecture 17 R T Howe Lecture 17 Last time Wrap up two port MOS amplifiers Today Sinusoidal signals Phasor representation of sinusoids Dept of EECS University of California Berkeley EECS 105 Spring 2002 Lecture 17 R T Howe Sinusoidal Function Review v t v cos t amplitude half of peak to peak Dept of EECS phase degrees or radians frequency radian 2 f 2 1 T University of California Berkeley EECS 105 Spring 2002 Lecture 17 R T Howe Graphical Description v 1 t v cos t v2 t v cos t 45 2 T Dept of EECS T University of California Berkeley EECS 105 Spring 2002 Lecture 17 R T Howe Why 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 leads to insights into the behavior of the circuit Dept of EECS University of California Berkeley EECS 105 Spring 2002 Lecture 17 R T Howe Linear 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 Dept of EECS University of California Berkeley EECS 105 Spring 2002 Lecture 17 R T Howe RC Circuit with Sinusoidal Input iR vs t vc t R iC C vs t Vs cos t set phase of source to zero use as the reference vc t Vc cos t solution is a sinusoidal signal with the same frequency but with a different amplitude and phase shifted with respect to the source Dept of EECS University of California Berkeley EECS 105 Spring 2002 Lecture 17 R T Howe A 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 j t d j t e j e dt Note that EEs use j 1 1 2 rather than i since the symbol i is already taken for current Dept of EECS University of California Berkeley EECS 105 Spring 2002 Lecture 17 R T Howe Using Imaginary Exponentials dvc RC vc vs dt vs t vs e j t Substitute vc t vc e j t Result j vc e Dept of EECS j t vc e j t vs e j t University of California Berkeley EECS 105 Spring 2002 Lecture 17 R T Howe Finding the Amplitude Ratio j vc e j e j Amplitude Ratio j j vc e e j e vc vs j t vs e j t use to find amplitude and phase j vc 1 e j j vs j e e 1 j Answer is a real number so take magnitude 1 vc 2 vs 1 Dept of EECS University of California Berkeley EECS 105 Spring 2002 Lecture 17 R T Howe Graphical Result for Amplitude Ratio 1 0 0 707 0 5 1 10 Dept of EECS 1 10 University of California Berkeley EECS 105 Spring 2002 Lecture 17 R T Howe Amplitude 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 Vc Vs dB 80 dB Vc Vs 0 707 Vc Vs dB 3 dB Dept of EECS University of California Berkeley EECS 105 Spring 2002 Lecture 17 R T Howe Finding the Phase j e j e j vs vc a real number Use Euler s formula to convert to rectangular form j cos j sin cos j sin vs vc Collect real and imaginary parts latter must be zero Im cos sin 0 tan Dept of EECS University of California Berkeley EECS 105 Spring 2002 Lecture 17 R T Howe Graphical Result for Phase 1 10 1 10 0 45 90 Dept of EECS University of California Berkeley EECS 105 Spring 2002 Lecture 17 R T Howe Finding 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 Re ve Dept of EECS j t v cos t University of California Berkeley EECS 105 Spring 2002 Lecture 17 R T Howe Pushing This Idea Further There are two parameters needed to define a sinusoidal signal magnitude phase Why not work with a complex number as the signal and eliminate the imaginary exponential from the analysis it cancelled out Define the complex number consisting of the amplitude and phase a sinusoidal signal as a phasor v t v cos t v t Ve j t V ve Dept of EECS j University of California Berkeley
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