Lecture University of California Berkeley by alicejenny

VIEWS: 3 PAGES: 26

									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
                                     105 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
                                               105 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 jL  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
                                  
                                               j0 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        j0
                                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  j0
                                                                                                         Q
                                              Q  100



                                         0
            Department of EECS                                                          University of California, Berkeley

								
To top