EECS 105 Fall 2003 Lecture 2 Lecture 3 Bode Plots Prof Niknejad Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 3 Prof A Niknejad Get to know your logs Engineers are very conservative A margin of 3dB is a factor of 2 power Knowing a few logs by memory can help you calculate logs of different ratios by employing properties of log For instance knowing that the ratio of 2 is 3 dB what s the ratio of 4 Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 3 Prof A Niknejad Bode Plot Overview Technique for estimating a complicated transfer function several poles and zeros quickly 1 j z1 1 j z 2 1 j zn H G0 j 1 j p 2 1 j p 2 1 j pm K Break frequencies 1 i i Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 3 Prof A Niknejad Summary of Individual Factors Simple Pole 0 dB 1 1 j Simple Zero 0 dB 1 1 90 90 1 j DC Zero 0 dB 90 j DC1Pole j Department of EECS 0 dB 90 University of California Berkeley EECS 105 Fall 2003 Lecture 3 Prof A Niknejad Example Consider the following transfer function 10 5 j 1 j 2 H j 1 j 1 1 j 3 1 100 ns 2 10 ns 3 100 ps Break frequencies invert time constants 1 10 Mrad s 2 100 Mrad s 3 10 Grad s j 1 j 5 10 2 H j 1 j 1 j 1 3 Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 3 Prof A Niknejad Breaking Down the Magnitude Recall log of products is sum of logs H j dB j 1 j 5 10 2 20 log 1 j 1 j 1 3 20 log j 20 log 1 j 105 2 20 log 1 j 20 log 1 j 1 3 Let s plot each factor separately and add them graphically Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 3 Prof A Niknejad Breaking Down the Phase Since a b a b 10 5 j 1 j 2 H j 1 j 1 1 j 3 j H j 1 j 5 10 2 1 j 1 j 1 3 Let s plot each factor separately and add them graphically Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 3 Prof A Niknejad Magnitude Bode Plot DC Zero 80 60 j 5 10 40 0 dB 20 104 105 106 107 108 109 1010 1011 20 40 60 80 Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 3 Prof A Niknejad Phase Bode Plot DC Zero 180 135 j 5 10 90 45 104 105 106 107 108 109 1010 1011 45 90 135 180 Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 3 Prof A Niknejad Magnitude Bode Plot Add First Pole 80 1 10 Mrad s j 105 60 dB 40 20 104 105 106 107 108 109 1010 1011 20 40 60 1 80 Department of EECS 1 j 7 10 dB University of California Berkeley EECS 105 Fall 2003 Lecture 3 Prof A Niknejad Phase Bode Plot Add First Pole 180 135 90 j 5 10 45 104 105 106 107 108 109 1010 1011 45 90 1 1 j 7 10 135 180 Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 3 Prof A Niknejad Magnitude Bode Plot Add 2nd Zero 80 2 100 Mrad s 1 j 8 10 60 dB 40 20 104 105 106 107 108 109 1010 1011 20 40 60 80 Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 3 Prof A Niknejad Phase Bode Plot Add 2nd Zero 180 1 j 8 10 135 90 45 104 105 106 107 108 109 1010 1011 45 90 135 180 Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 3 Prof A Niknejad Magnitude Bode Plot Add 2nd Pole 80 60 3 10 Grad s 40 20 104 105 106 107 108 109 1010 1011 20 1 40 1 j 10 10 60 dB 80 Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 3 Prof A Niknejad Phase Bode Plot Add 2nd Pole 180 135 90 45 104 105 106 107 108 109 1010 1011 45 90 1 j 135 1010 180 Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 3 Prof A Niknejad Comparison to Actual Mag Plot Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 3 Prof A Niknejad Comparison to Actual Phase Plot Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 3 Prof A Niknejad Why do I say actual I plotted the transfer characteristics with Mathematica The range of frequency for the plot is 6 orders of magnitude The program has to find the hot spots in order to plot the function Near the hot spots more points are plotted In between hot spots the function is interpolated If you pick the wrong points you ll end up with the wrong plot mag LogLinearPlot 20 Log 10 Abs H x x 10 4 10 11 PlotPoints 10000 Frame True PlotStyle Thickness 005 ImageSize 600 GridLines Automatic PlotRange 10 4 10 11 20 100 Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 3 Prof A Niknejad Don t always believe a computer Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 3 Prof A Niknejad Second Order Transfer Function The series resonant circuit is one of the most important elementary circuits The physics describes not only physical LCR circuits but also approximates mechanical resonance mass spring pendulum molecular resonance microwave cavities transmission lines buildings bridges Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 3 Prof A Niknejad Series LCR Analysis With phasor analysis this circuit is readily analyzed Vo Vs I Vs I Department of EECS 1 j L I I R j C 1 j L R j C Vs V0 I R R 1 j L R j C University of California Berkeley EECS 105 Fall 2003 Lecture 3 Prof A Niknejad Second Order Transfer Function So we have Vo V0 R H j Vs j L 1 R j C To find the poles zeros let s put the H in canonical form V0 j CR H j Vs 1 2 LC j RC One zero at DC frequency can t conduct DC due to capacitor Department of EECS University of California Berkeley EECS 105 Fall 2003 Lecture 3 Prof A Niknejad Poles of 2nd Order Transfer Function Denominator is a quadratic polynomial R j L V0 j CR H j 2 1 R Vs 1 LC j RC 2 j j LC L R j 1 2 L H j 0 R LC 02 j 2 j L j H j Department of EECS 0 Q 0 j j Q …
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