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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! dB ratio dB ratio -20 0.100 20 10.000 -10 0.316 10 3.162 -5 0.562 5 1.778 -3 0.708 3 1.413 -2 0.794 2 1.259 -1 0.891 1 1.122 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 )K (1 j p 2 )(1 j p 2 ) (1 j pm ) 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 1 1 Simple Pole: 0 dB 1 90 1 j Simple Zero: 0 dB 90 1 j DC Zero: 0 dB 90 j DC Pole: 0 dB 1 90 j Department of EECS University of California, Berkeley EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad Example Consider the following transfer function 105 j (1 j 2 ) 1 100 ns H ( j ) (1 j 1 )(1 j 3 ) 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 ) 10 5 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 j (1 j ) 105 2 H ( j ) dB 20 log (1 j )(1 j ) 1 3 j 20 log 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 105 j (1 j 2 ) H ( j ) (1 j 1 )(1 j 3 ) j H ( j ) 5 1 j 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 j 60 5 10 40 20 0 dB 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 j 180 5 135 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 60 10 5 dB 40 20 104 105 106 107 108 109 1010 1011 -20 -40 -60 1 -80 1 j Department of EECS 107 dB University of California, Berkeley EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad Phase Bode Plot: Add First Pole 180 j 135 5 10 90 45 104 105 106 107 108 109 1010 1011 -45 -90 1 -135 1 j -180 107 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 60 10 8 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 180 135 1 j 108 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 -60 1010 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 − 1 Vs I jL I IR j C 1 Vs I j L R j C Vs V0 I R R 1 j L R j C Department of EECS University of California, Berkeley EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad Second Order Transfer Function So we have: + V0 R Vo H ( j ) Vs 1 − j L 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 V0 j CR H ( j ) L Vs 1 LC j RC 2 1 ( j ) 2 j R LC L R j 1 H ( j ) L 02 R LC 02 ( j ) 2 j L 0 j Q 0 L H ( j ) Q 0 ( j ) j 2 0 2 R Q Department of EECS University of California, Berkeley EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad Finding the poles… Let’s factor the denominator: 0 ( j ) j 2 0 0 2 Q 0 02 01 j0 1 2 4Q 0 2Q 4Q 2Q Poles are complex conjugate frequencies Im The Q parameter is called the “quality-factor” or Q-factor Re This parameters is an important parameter: Q R 0 Department of EECS University of California, Berkeley EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad Resonance without Loss The transfer function can parameterized in terms of loss. First, take the lossless case, R=0: Im 02 Re 0 2 0 j0 2Q 4Q Q When the circuit is lossless, the poles are at real frequencies, so the transfer function blows up! At this resonance frequency, the circuit has zero imaginary impedance Even if we set the source equal to zero, the circuit can have a steady-state response Department of EECS University of California, Berkeley EECS 105 Fall 2003, Lecture 3 Prof. A. Niknejad Magnitude Response The response peakiness depends on Q 0 R j j 0 0 L Q H ( j ) 0 R 0 0 j 2 2 0 j 2 2 0 L Q H ( j 0 ) 1 Q 1 02 j Q H ( j 0 ) 1 H (0) 0 0 Q 10 02 02 j0 Q Q 100 0 Department of EECS University of California, Berkeley