# Lecture University of California Berkeley by alicejenny

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```									EECS 105 Fall 2003, Lecture 2

Lecture 3: Bode Plots

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

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