EE105 Fall 2005 Microelectronic Devices and Circuits Lecture 17 Frequency Domain Analysis Announcements Homework 7 due today Homework 8 due next week Lab 6 this week Reading Chapter 10 10 1 2 1 Lecture Material Last lecture Common drain amplifier Review phasors This lecture Frequency domain analysis Bode plots 3 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 4 2 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 d e j t j e j t dt Note that EEs use j 1 1 2 rather than i since the symbol i is already taken for current 5 Using Imaginary Exponentials dv RC c v c v s dt Substitute Result v s t v s e j t v c t v c e j t j v c e j t v c e j t v s e j t 6 3 Finding the Amplitude Ratio j v c e j v c e j e j t v s e j t j e j e j v c v s use to find amplitude and phase vc 1 e j v s j e j e j 1 j Amplitude Ratio Answer is a real number so take magnitude vc 1 vs 1 2 7 Graphical Result for Amplitude Ratio 1 0 0 707 0 5 1 10 1 10 8 4 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 9 Finding the Phase j e j e j v s v c a real number Use Euler s formula to convert to rectangular form j cos j sin cos j sin v s v c Collect real and imaginary parts latter must be zero Im cos sin 0 tan 10 5 Graphical Result for Phase 1 10 1 10 0 45 90 11 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 j t v cos t 12 6 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 cancels 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 j 13 Using Phasors Capacitor Current ic t C vc t ic t C ic t Ic e j t dv c dt v c t Vc e j t Result 14 7 Impedance of a Capacitor Definition the impedance Z of a two terminal circuit element is the ratio of the phasor voltage to the phasor current positive reference convention Ic C Vc Zc Admittance Yc 1 Zc 15 Using Phasors Inductor Voltage iL t di v L t L L dt L vL t v L t VLe j t i L t IL e j t Result 16 8 Inductor Impedance IL VL L ZL Admittance YL 1 ZL 17 Circuit Analysis with Phasors Assumption sources are sinusoidal steady state R vs t C v t c Ratio of output to input phasor is the transfer function of the circuit V H c Vs 18 9 Bode Plots 1 Plot magnitude H in dB vs log scale 2 Plot phase H in degrees vs log scale 19 Bode Plots for Low Pass Filter 1 Plot magnitude H in dB vs log scale 2 Plot phase H in degrees vs log scale H dB 1 1 dB 1 j 1 j dB 20 10 Sketching the Magnitude Plot 1 H dB 1 j 1 20 log 2 dB 1 Low frequency 1 asymptote High frequency 1 asymptote 21 The Break Frequency 3dB 1 H dB 1 10 1 10 100 20 40 60 22 11 Finding the Phase Plot 1 H 0 arctan 1 j Why Low frequency asymptote High frequency asymptote Approx linear with for 1 10 10 23 Rapidly Sketching the Phase Plot Phase H 1 10 1 10 100 45 90 24 12 The High Pass Filter vs t C R vr t Vs 1 j C R Vr V H r Vs 25 Magnitude Bode Plot H dB j 1 dB j dB dB 1 j 1 j First term numerator 26 13 Graphical Addition of Magnitudes H dB 40 20 1 10 1 10 100 20 40 27 Phase Bode Plot for HPF Phase H 90 1 10 1 10 100 90 28 14
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